I had a 2001 Suburban with the 5.3 and the 4.10 axles. I pulled three different trailers with it on long road trips.
Trailer 1: 29' Trail Cruiser, ~5000 lbs, three trips from MI to Florida
Trailer 2: Captiva 28'(?), ~5500 lbs, one trip from MI to Washington DC and Williamsburg VA
Trailer 3: Salem 26', ~6500 lbs, one trip from MI to Yellowstone and Colorado
All 3 rigs used a traditional hitch setup - WD hitch with friction sway brake.
I had zero trailer experience, but researched the hell out of weights, lengths, sway, etc.
All 3 trailers exhibited moderate push-pull when passing/getting passed by semis.
On one trip to Florida, winds were gusting to 40-50 MPH. In hindsight, we probably just shouldn't have traveled at all that one day. I had the friction brake cranked down so hard that I bent the 1/2" plate that the vertical sway ball was attached to on the hitch.
In all of those trips, spanning 15-20,000 miles of towing, I never had any situation where sway was uncontrollable or made me feel like I was going to lose it.
The 3.73 axle won't have you winning any races. I always towed in 3, never in "D." At 65 MPH, the engine was around 3000-3100 RPM. With the 3.73, you should be a little lower.
Driving through the Rockies sucked. Climbing Powder River Pass in Wyoming (9666 ft), the best the truck could as we approached the pass was 25 MPH in first gear.
I also went through a few rear ends. First, I blew the pinion bearing because I was trying to use downshifts to help slow the rig. Next, I fried the rear end by driving 500 miles across MN and SD on a 95° day at 65-70 MPH. My advice to you is to purchase a handheld infrared thermometer, so you can keep an eye on the temperature of the rear axle on long drive days. Buying an aftermarket diff cover with cooling fins would probably be a good idea, too. The rear axle is definitely the weak link in these trucks.
Feel free to PM me if you have any more questions.
PS - note the truck in my sig. If you plan to use your trailer frequently, an upgrade to a 2500 is like night and day.
Trail Cruiser:
Captiva:
Salem:
2008 Suburban 2500 3LT 3.73 4X4 "The Beast"
2013 Springdale 303BHS, 8620 lbs
2009 GMC Sierra 1500 Denali (backup TV, hot rod)
2016 Jeep JKU Sahara in Tank, 3.23 (hers)
2010 Jeep JKU Sahara in Mango Tango PC, 3.73 (his)