Is that 6000 lbs the actual loaded gross weight, or perhaps the Unladen Weight from a sticker or brochure? And how many people inside the tow vehicle? Cause a 5.3L V8 on a Burb ought to manage 6000 lbs ok. Maybe not swiftly, but ok. Of course, I don't know what you consider to be "barely". It's going to slow down a bit.
Does the Burb have the 3.73 "economy" rear axle? That gives it a tow rating around 7000 lbs, so with passengers and gear it is near max.
In any case, a half ton pick-up equipped to tow 9000 lbs or so should be an improvement. Get a GM or Ford Tow Guide for the years you are interested in and check out which vehicles can do the job. Major factors are weight carried in the vehicle, cab style, engine size and rear axle ratio.
https://www.xr793.com/wp-content/uploads/2017/07/2008-GMC-Trailering-Guide.pdf
https://www.fleet.ford.com/towing-guides/