A 350 bull is a pretty tall order in any unit in Utah now days, especially given that we have been in a prolonged drought, and appear to be headed for more.
I had a tag in one of the top units in the state last year, and while I saw a ton of bulls in 6 days of hunting, only a couple were in your goal range.
That said, I had a hunt of a lifetime with screaming bulls in my face every day. We hunted hard and experienced everything I had dreamed of with an elk rut hunt. I came home with a very nice bull, but the inches of antlers I took had little to do with how amazing my hunt was for me.
My advice is, if you do draw, go to hunt hard, experience as much rut activity as you can and just have a blast, without worrying about the size of bull you bring home.
Good luck in the draws.