I have the previous generation Suburban as my tow vehicle, C2500 with the 7.4L. For its age and vintage, it pulls my 32' trailer with relative ease and little sway. I use a WDH system with the spring bar / chain setup and friction type anti-sway (also older design) that both help, as well as Airlift airbags on the rear suspension of the Suburban that stabilize the ride even more. Any 3/4 ton vehicle will be more stable than its 1/2 ton counterpart. Not only are the engine/transmission combo's bigger and stronger, but it also has beefier suspension, brakes, tires, and engine cooling systems. All of those factors will improve your towing situation. The 8.1L Suburban you are considering (assuming it has the 4:10 gears... RPO code GT5 on the sticker inside the glovebox) has a 12,000# towing rating.
That said, in general towing a large trailer DOES take more "work" from the driver than driving a car. There is more to think about, more to watch, new feelings and sway, etc. that really tire you out quicker. Part of what you may be experiencing is, well... inexperience. You're brain and body will get more used to the feel of pulling the trailer after a few more trips, so you might want to give that a try (and make sure your hitch etc. is properly adjusted) before dropping money on a different tow vehicle.