You have two main issues you'll be fighting: holiday campers, and reduced campsites due to lots of snow still at higher elevations.

My first choice, if I were you, would be to look towards the High Uintas. Most of the lower campgrounds are open already, so you might have some luck there. There are also some areas along the northern slope of the Uintas, getting into Wyoming, but I'm not as familiar with those. I've been to the Henry's Fork area, and on those forest roads, but they're probably still snowpacked.

I don't know the Wasatch Mountain areas north of SLC, so there might be some options up there as well.