Towing with a Tahoe

I'll state up front that I have not towed with a Tahoe, just a 2500 Suburban. If you are comparing a 1500 Suburban to a Tahoe, I'd think they would have similar stability. The Tahoe has a shorter wheelbase, but it also has a shorter rear overhang which should balance it out. Even on my 2500 Suburban, tires make a huge difference. I towed our trailer (28.5' ball to bumper, 6k empty, 7500+ loaded with a scaled 1200-1300 lb tongue weight) for 4 years with Michelin all season load range E tires and it was incredibly stable. Just putting new tires on that were all terrain definitely reduced the stability. I didn't really get blown around so much but when I had to make a steering correction the was definitely less precise. Just overall felt less stable and secure. We did tow it 2-3,000 miles this year and it was fine, but it made it obvious to me how different tires can behave. If you don't need all terrain tires, get an all season tire with a more stable tread design. If you can, get LT tires for it. The rear suspension has a Panhard rod that should keep the rear axle pretty stable. The weakest link might be tires if there is an issue.

Porpoising could possibly be cured with good shocks. I've always preferred Bilstein shocks, not just because they work well, but because they last a long time.

Good luck. You might consider weighing your current set-up to make sure you know what you're dealing with. Can't hurt.