I had an 06 tahoe with the same 5.3 and NEVER had an issue towing. I hated to get rid of it, but needed the room for my 4th child. Now I have an Yukon XL. I have not towed with it yet, but would assume that it tows similar.
Sounds like you got a lemon from the get go. Sorry bout that.
As far as advice for a new vehicle....you cant beat the 5.3 {almost cant}.
Or like many others have pointed out, find something with the 6.0. They tow like crazy and always have plenty of power when you need it.
Good luck with your next purchase.