>>2311727OP here, I'm in northern UT around Bear Lake. Lived in Jackson for a decade saving money and just bought a place in the middle of nowhere.
Not too familiar with the sawtooths unfortunately, but with how dry this Winter has been, feel like you could probably pull it off by late June and avoid the Summer crowd/ fire season.
A buddy of mine told me there is supposed to be a pretty cool trail in City of Rocks that takes 2-3 days, but he doesn't remember the name and I can't find any solid info online.