Assuming you got at least the 5.3 V-8, the FC 28 is within your towing spec. The bigger question is the truck's payload. That will be listed Ina yellow and red sticker on the driver's side door frame. For the typical Silverado crew cab pickup, that will be in the vicinity of 1500
lbs. unless you have the hard-to-find max trailer tow package, in which case it will be over 1900 lbs. If you have a "LT" model without a sunroof, it will be a bit more. If memory serves, Airstream says an FC 28 has a nominal tongue weight of 800 lbs. with full propane bottles, batteries. People on this forum report somewhat higher--around 1000 lbs. based on scale readings. So your truck is carrying that tongue weight plus the weight of all the people, dogs and stuff inside the vehicle. As you can see, you're pretty close to the limit. I spent the better part of a year shopping for a TV after I had bought an FC 27. I finally ended up with a '15 Sierra with the 6.2 engine and the max trailer tow package. This package has a different rear axle and a beefier rear suspension than standard and works well (although I have not tackled the Rocky Mountains yet).
I would suggest you invest in a good weight distributing hitch. This, when properly set up, will shift several 100s of pounds of tongue weight off of the truck and back to the trailer. The ProPride (and similar HENSLEY) also have anti-sway geometry (not just friction) that, in my own experience, is 100% effective at eliminating sway, even when you're passed by a semi in the adjacent lane going 10 mph faster. These hitches are more expensive than most, are not as easy to connect to and turn a little differently--so they have their drawbacks. But given that you are probably loading your truck at 100% of rated cargo, or more, it would be nice to avoid the sway problem, which can lead to a loss of control.
If your travels are limited to Texas and other plains states, I'm sure your drivetrain will be fine, so long as you observe the 60 mph speed limit imposed by the Goodyear Marathons on your trailer. If the Rockies are in your travel plans, you might have a look at added cooling for the engine, transmission and lube oil.