Berthoud Pass, wolf Creek Pass, Marble, and Red Mountain Pass in Colorado are choice spots in Colorado. Actually, more accessible bc riding in terrain in the state than just about anywhere.
The Cottonwoods in Utah are great. The BC riding off of Ben Lomond above Ogden is another good spot.
Teton Pass in Wyoming is stellar.
California has a lot of great stuff around Donner and Carson Spur. Plus whatever that pass is from South Lake to Carson City. More stable snow than what is found in Colorado or even Utah. If they are in a storm cycle. It can be long periods between storms.
Washington is another spot to look at. Some of the most crazy backcountry terrain you can ride actually. Utah, Colorado, even California don't really have terrain like that. Of course as great as it can be, it can rain just as much. That is a drawback.
If you have 2-4 weeks, I'd say loop around from Colorado to Utah and up to Wyoming. You could do the Front Range Berthoud Pass to Marble outside of Aspen tour, or go the Southern route and hit Wolf Creek and Red Mountain pass out of Silverton. Then drive up to the Cottonwoods and hit that great stuff up and finish off at Teton Pass for a few days.
Cheap places to stay are found at most of the places with maybe the exception of Teton Pass. I think the hostel in Jackson is reasonable. That would be a hell of a road trip and you'd get to ride some of the best backcountry spots around.
Avy knowledge is key for sure. Pretty much all of the places I've mentioned are pretty easy to slum for partners who are knowledgeable of the terrain at the trail heads. Berthoud, Cottonwoods, Red Mountain probably being the easiest. Shouldn't be hard at Wolf Creek either. As long as you are at the Summit by 9am. I would think Teton Pass wouldn't be hard, but there are some issues there, that may make people not as friendly as at the other spots mentioned. Resource issues making locals a bit more jaded is all. Still, most everyone I've met in the backcountry is in a good mood for a good reason.