welcome to the forum.
I've also towed with several 3/4 tons and and several one ton DRW trucks. Non of my drw trucks was any more stable pulling the same size trailer my 3/4 ton trucks could pull. I bought the DRW to pull heavy 15k-20k lbs plus trailers that had more hitch weight than a 3/4 ton truck could carry.
The F250 truck chassis will handle a 11xxx lb trailer just fine. The F350 srw is the same truck but has a heavier rear spring pack. In fact if the F250 you choose has the heavy service pack or camper pack it has the same suspension as the 350 srw truck.
And if you choose a F350 srw be aware some come with the same 10k gvwr as the F250. Get the 11500 gvwr with 7230 rawr.
You don't need a one ton drw to pull a small trailer that weight.
Gas or diesel ?? Your choice there.