There are a number of ratings for a TV and you need to meet all of them. With a 5.3 engine I’m guessing you’re talking about a 1/2 ton and the weakest link with them is payload. Tow rating is almost a pie in the sky rating. The best way to have any reliable numbers is to load up the TV with all the people and gear you’ll carry on a trip and weight it. Then subtract that weight from the GVWR on your door post for your real payload. If you’re well under that you’ll be fine, if you’re under but pushing it, it will be OK on the flats but not so if you get in the mountains. People pulling near max usually wind up getting a bigger TV .
I don’t pull a long rig so I defer comments about pulling that length with a ½ ton to people who do. But a 6000# dry 34’ TT will likely be well over 7000# on the road and it’s too much TT for the TV to me.