Check out the following website for road conditions in the White Mountains:
https://www.az511.gov/map
At present, it looks like Hwy 191 is open without any restrictions. Hwy's 273 and 261 are still closed due to the snow pack. I saw a recent photo from the Big Lake area and it appears there is still about 3' - 4' of snow on the ground, access only made via snowmobile or tracked vehicle. I highly doubt the Three Forks Road will open anytime soon, as well.
Regarding where to stay, several of us stayed at the Sportsman's Lodge in Alpine. Nice, comfy beds and rooms. The owners are riders who obligingly offer used towels, buckets and soap to wash your bike when you arrive.
https://www.sportsman778.com/ for more information on this motel. Dinner, breakfast or lunch is a short walk to either Bear Wallow Cafe or the cafe across the street (sorry, can't recall the name). Both are good.
I'm looking forward to doing this ride soon as well, staying in the Granville CG once again just north of Morenci. Last year saw a group of us pulling into here just after dark, so we didn't really have a chance to enjoy the campground.
- David