Unfortunately you are misinterpreting quite a few items. Here they are:
1. There is no such thing as GVR of the trailer, so I think the sticker states GVWR (or GTWR for trailers) or the "gross vehicle weight rating". This is the MAXIMUM weight the trailer is allowed to weigh with all the gear, propane, AND any tanks that are full. The trailer should never be allowed to weigh more than the GVWR. This limitation is calculated based on the frame, suspension, axles, wheels, tires, brakes and shocks (if included). Any food, water, dishes, luggage, clothing, pots, pans, games, coolers, EVERYTHING you put in there must be included in this maximum weight! What you do is put all the gear in the trailer, have it weighed (along with the truck) and hopefully the GVW (gross vehicle weight) of the trailer is LESS than the GVWR (GVW Rating) to stay within this rating (limit). Hopefully the other numbers look good too. If you don't know what numbers I'm talking about, please ask!
2. The WDH does NOT increase the GVWR of the Tahoe! It only helps shift some of the weight of the towed item (trailer in your case) to be carried by the vehicle's frame and front axle, instead of ALL the tongue weight on the rear axle thereby making for an unsafe ride. If the Tahoe's hitch only has a rating of 500 lbs, then the maximum the truck can tow is a 5,000 lb trailer assuming that 10% of that weight is carried on the ball (or 500 lbs). The trailer you describe is too heavy for the Tahoe since it will have 625 lbs (or more, depending on how the trailer is loaded).
3. Sway control is not standard with a WDH. You must buy a WDH that has one, or you add a sway control to a WDH that does not have one. This is a conscious decision on which option you seek. The integrated WDH/Sway control hitch system is more expensive than a WDH with a separate and additional Sway control bar that needs some installation. I bought my hitch for about $200, and the sway bar for $80. These are pretty good prices because I got them from the dealer because I bought the trailer from them too. The dealer included installation and adjustment for free which tells me I paid too high a price for my trailer (ahem).
4. Just because the tires are worn does not mean they are bad. If the trailer's tires are more than 5 years old, you should replace them because their lack of use increases the chances of tire rot which you CANNOT see until it's almost too late even it there's still a lot of tread. Most vehicle tires are used so much they rarely go beyond the 5 years and get replaced more than likely from wear and tear than sitting for weeks or months at a time like on a trailer. For a trailer, do the "head of the coin" test or check the DOT code (ask for help it you don't know about this) for more than 5 years old. The Tahoe is probably okay and probably only needs the "head of the coin" test. Just remember, tires that sit for long periods of time will dry rot and this shortens their life significantly.
5. The Chevy Tahoe may have a higher tow rating than the 5,000/500 limits of the hitch. If was not unusual for dealers to make a vehicle "tow ready" and not put all the best stuff in it. I'll see it I can find the tow ratings for your vehicle, then see if the truck was not "short-changed" will lesser rated stuff. For example, I had a late model Ford Expedition with an 8,500 lbs tow rating and Class IV receiver. But when I bought my 8,000 lb trailer, even though I ended up with a 9,000 lb WDH, I still could only tow a maximum of 8,500 lbs because it was other components of the truck that was limited.
6. You must also know the Combined Gross Vehicle Weight Rating (CGVWR) for the Tahoe. This is the weight of BOTH the Tahoe and the trailer combined. This covers not loading the truck up to the max, then loading up the trailer to its max, too and expect to tow, drive and stop all this weight up and down hills, and at speed.
7. FYI - The maximum a truck can tow is limited by something that you may not know about. DO NOT do what so many have done - bigger tires and wheels, air bags, beefier hitch and thinking now you can tow more. The thing you beef up may not be the ONLY item that is limiting the weight rating. Other items will include: engine, transmission, drive shaft, U-joints, wheels, tires, differential, brakes, shocks, springs, suspension parts, frame, and so many other things.
8. DO NOT allow a salesman to sell you what you think you need. Half the time they are wrong about what they are selling and don't even know it. I was told by a Ford service person, a Ford Sales person and by a suspension shop technician that air bags would solve a weight problem. This is NOT true. They thought that because the air bags removed the vehicle's back end sag, this means I could tow more weight. But this is NOT true. Look at the fine print of the owner's guides of these aftermarket parts and they will state this is NOT so.
I hope I provided enough to start with. You might not be in weight trouble yet unless you stop immediately, then start checking, confirming, weighing and researching what you have and what you need. If ANY part of your tow vehicle and trailer are over the weight ratings, then the rig combination is overweight, period! You WILL be liable for any mishap if other property or lives are affected.
1. There is no such thing as GVR of the trailer, so I think the sticker states GVWR (or GTWR for trailers) or the "gross vehicle weight rating". This is the MAXIMUM weight the trailer is allowed to weigh with all the gear, propane, AND any tanks that are full. The trailer should never be allowed to weigh more than the GVWR. This limitation is calculated based on the frame, suspension, axles, wheels, tires, brakes and shocks (if included). Any food, water, dishes, luggage, clothing, pots, pans, games, coolers, EVERYTHING you put in there must be included in this maximum weight! What you do is put all the gear in the trailer, have it weighed (along with the truck) and hopefully the GVW (gross vehicle weight) of the trailer is LESS than the GVWR (GVW Rating) to stay within this rating (limit). Hopefully the other numbers look good too. If you don't know what numbers I'm talking about, please ask!
2. The WDH does NOT increase the GVWR of the Tahoe! It only helps shift some of the weight of the towed item (trailer in your case) to be carried by the vehicle's frame and front axle, instead of ALL the tongue weight on the rear axle thereby making for an unsafe ride. If the Tahoe's hitch only has a rating of 500 lbs, then the maximum the truck can tow is a 5,000 lb trailer assuming that 10% of that weight is carried on the ball (or 500 lbs). The trailer you describe is too heavy for the Tahoe since it will have 625 lbs (or more, depending on how the trailer is loaded).
3. Sway control is not standard with a WDH. You must buy a WDH that has one, or you add a sway control to a WDH that does not have one. This is a conscious decision on which option you seek. The integrated WDH/Sway control hitch system is more expensive than a WDH with a separate and additional Sway control bar that needs some installation. I bought my hitch for about $200, and the sway bar for $80. These are pretty good prices because I got them from the dealer because I bought the trailer from them too. The dealer included installation and adjustment for free which tells me I paid too high a price for my trailer (ahem).
4. Just because the tires are worn does not mean they are bad. If the trailer's tires are more than 5 years old, you should replace them because their lack of use increases the chances of tire rot which you CANNOT see until it's almost too late even it there's still a lot of tread. Most vehicle tires are used so much they rarely go beyond the 5 years and get replaced more than likely from wear and tear than sitting for weeks or months at a time like on a trailer. For a trailer, do the "head of the coin" test or check the DOT code (ask for help it you don't know about this) for more than 5 years old. The Tahoe is probably okay and probably only needs the "head of the coin" test. Just remember, tires that sit for long periods of time will dry rot and this shortens their life significantly.
5. The Chevy Tahoe may have a higher tow rating than the 5,000/500 limits of the hitch. If was not unusual for dealers to make a vehicle "tow ready" and not put all the best stuff in it. I'll see it I can find the tow ratings for your vehicle, then see if the truck was not "short-changed" will lesser rated stuff. For example, I had a late model Ford Expedition with an 8,500 lbs tow rating and Class IV receiver. But when I bought my 8,000 lb trailer, even though I ended up with a 9,000 lb WDH, I still could only tow a maximum of 8,500 lbs because it was other components of the truck that was limited.
6. You must also know the Combined Gross Vehicle Weight Rating (CGVWR) for the Tahoe. This is the weight of BOTH the Tahoe and the trailer combined. This covers not loading the truck up to the max, then loading up the trailer to its max, too and expect to tow, drive and stop all this weight up and down hills, and at speed.
7. FYI - The maximum a truck can tow is limited by something that you may not know about. DO NOT do what so many have done - bigger tires and wheels, air bags, beefier hitch and thinking now you can tow more. The thing you beef up may not be the ONLY item that is limiting the weight rating. Other items will include: engine, transmission, drive shaft, U-joints, wheels, tires, differential, brakes, shocks, springs, suspension parts, frame, and so many other things.
8. DO NOT allow a salesman to sell you what you think you need. Half the time they are wrong about what they are selling and don't even know it. I was told by a Ford service person, a Ford Sales person and by a suspension shop technician that air bags would solve a weight problem. This is NOT true. They thought that because the air bags removed the vehicle's back end sag, this means I could tow more weight. But this is NOT true. Look at the fine print of the owner's guides of these aftermarket parts and they will state this is NOT so.
I hope I provided enough to start with. You might not be in weight trouble yet unless you stop immediately, then start checking, confirming, weighing and researching what you have and what you need. If ANY part of your tow vehicle and trailer are over the weight ratings, then the rig combination is overweight, period! You WILL be liable for any mishap if other property or lives are affected.