Air Bags are a good idea, they will help the sagging problem you have, (with any truck), they can hold up to 5000lbs capacity, if thats is still not enough then you can go with a timbren kit that can handle up to 8000lbs, i personally went with a timbren kit cause when I wasn't towing or hauling anything I did not lose my factory ride or have to worry about the bags ripping or worry about filling the bags every time I wanted to tow something, i have the same truck as you different year tho, this was the kit i got i put it in and forgot about it!!!!! which was the best part.
GMRCK25S Timbren Kit 1999-2010 Silverado 2500HD Rear Timbren Springs | $199.63

I would go with the timbren kit as well if you were to keep your current truck. But if you were to go with your other option with the 2500Hd, you would be set with just the 6.0. My 2010 pulls great, i pull a 24ft vnose enclosed trailer for work and it pulls quiet nice. Its got the power that you need and will handle the weight nicely.

I tow with a '04 Silverado 5.3/3.73 setup most every day. I tow a 4K work trailer and a 7+K dump/trash trailer. No problem with either. The truck has a "redneck" ram air setup, Flowmaster cat-back, Hellwig air bag setup (worth every penny) and most importantly, a Blackbear Performance scan tune. Truck tows anything I've hooked to it, obviously within reason but 6-7K is no problem at all. 9K starts to feel heavy and I don't go much over that.

These trucks have long leaf springs. Long springs can still have good load carrying capacity but they also have a long range of travel and so they sag more that you would expect. The Hellwig setup fixes that completely.

I've checked trans temp a few times after towing hard with an IR thermometer and the trans was never that hot so I did not bother with a trans cooler. I did go to Amsoil synthetic fluid.

I think your truck is perfect for what you need with some easy upgrades. If you towed every day then I would say maybe the 3/4ton would make you more comfortable. But for weekends, etc. what you have is fine. Just upgrade some bits and you're good to go.

Regarding adding air to the Hellwig setup, its not a big deal. I have a mini compressor that is always around. Get a nice digital tire gauge too. Air up before you tow/haul. Air down when not. Once a week is no big deal, I adjust mine 2-3 times a week, it takes just a minute or two.

RT