I have spent a fair amount of time in the Price area over the years. Skyline drive would certainly be a **** shoot that early and it will totally depend on snowpack and/or your vehicle choice atv/utv/suv/truck. The biggest problem with Skyline that time of year is the drifts. You may go for miles on dry road and then hit a snow drift. So if you can make sure and take a shovel sometimes it only takes a little bit of work to get by other times it may take a grader You will also potentially run into drifts on the access roads to and from for example in the Joes Valley area there is a major drift area not to far from top just before you get to Skyline Drive. Obviously you can go back the way you came just make sure you have enough fuel etc to do that and not plan on getting down at point b and filling up and taking the highway back to your starting point. However for the most part there are roads at lower elevation that more or less parallel Skyline (they dont go all the way to 70 but down to Ferron or so. The San Rafael Swell should be good to go by then unless like previously mentioned a recent heavy rainstorm. Unfamiliar with the Duchesne area. IF you are willing to go just south of 70 you still have the Swell, Goblin Valley area and the Green River Desert.
From where hwy10 and 70 intersect you can go south over the mountains on 72 the road while paved and is decent road is a beautiful road that will end you up in the Fremont, Bicknell area just west of Capital Reef NP. Once east of that you have Canyonlands NP Maze district. One of the more remote places in the contiguous 48 with a few roads in it( We did from Hans Flat Ranger Station to Hite approx 100 miles in my truck no problem the Flint trail has several switch backs that required backing up to make but at least at the time was not a big deal not like some of the hardcore stuff in the Moab area. If you plan on camping in there you better have a reservation as there are very few campsites but they are awesome particularly the Golden Stairs site. However you can get in and out in a day just depends on what all you want to see and do in there.
Sorry for the rambling post love it out there, been going for over 50 years there will be plenty of good areas but the higher elevation stuff will totally depend on snowpack