You might want to double-check your payload on the door sticker. I doubt it's 2500. Probably closer to 2000 or 2100. Payload of my 2008 is 2088 lbs.
Yes, you will be fine with that trailer. Frankly, GM underrates the 2500 trucks, so even if you're close to the limits, you still have a safety margin.
What you definitely will want to do is check your weights on a truck scale once you're loaded and ready for your first trip. Good confirmation to make sure you're properly loaded and balanced, and not overloading anything.
Airbags are not necessary. The leaf springs are plenty stout. You will likely want to upgrade your shocks. If you have the stock suspension, Bilstein 4600s will make a HUGE difference when towing.
I've towed my 8600-lb trailer all over the country, including some Rocky Mountain passes and the Eisenhower Tunnel. She does just fine. I'm never a menace causing slowdowns behind me, and I'm always faster than the big rigs. The 6.0 loves to rev - let it eat!
Our rig parked at Wall Drug. Notice no WD on the hitch, yet she still sits perfectly level. 1120-lb tongue weight.