If you budget $20k for the TT, I'd say get a $10k used TT, take the $15k value of the current Burb, and buy a $25k 3/4 ton that is lower mileage, maybe get into a 2008+. That will go a lot farther towards towing comfort than spending $3k on your current Burb.
If you insist, flush all the fluids. Spring for a WDH with integrated sway control over the basic design. Replace the shocks with HD towing ones. At 13 years old, it may need new springs or replacement air ride system as some of them had. You can change the gears to 4.10 or 4.56. You can supercharge that 5.3L. Get some OEM towing mirror clones from Ebay. If you have the OEM round tube receiver, get the Curt aftermarket. Endless way to blow $10k on your current TV to still not equal a stock 8.1L well maintained 3/4 ton Burb.