Now that you've determined your carrying capacity, let's look at how your truck might perform towing 6 to 7K lbs.
These HP/Tq curves will give you some idea of what RPM the 5.3L will run at to move that load.
My 8.1L runs at about 2200 RPM to tow a 7200lb TT at 62mph on a flat road. Looking at the HP curve, it is outputting 180HP and 440lb-ft TQ.
Under the same conditions, the 5.3L will need 180+HP to maintain 62mph. On that curve, the RPM at 180HP is 2800RPM. But the TQ is at 350lb-ft so not as much work is getting done. So the 5.3 can maintain 62mph at say 2800 to 3000RPM, the rise and fall of the road grade will cause the speed of the engine to vary greatly to maintain 62mph. Cruise control may not work very well if the transmission is constantly shifting up and down.
Bottom line is: Expect to be running between 2800 and 3800RPM while towing. Much higher in the mountains. That engine will run all day at those RPMs so let er sing!

