My Dodge 2500 came with airbags. They have a REALLY slow leak at the switch that I've not gotten around to fixing. I have a deck, two heavy sleds (~650 & ~750, respectively), plus a bunch of crap in the bed.
With the airbags and 40ishpsi, no problem.
They bled down one day, and it was cold enough that something in my compressor/tank froze - could not refill.
It was not SCARY, but the additional sway made me nervous.
Doable, yes. Heck, about what you'd expect putting 1200# on the back of your truck.
WITH airbags, though, I barely notice the sleds up there.
So...doable, sure, absolutely, but you'll feel them flopping around occasionally. I'd put airbags on, personally, but had I not gotten used to it WITH airbags, I'd probably not mind without.