Second question first: Most hotels/motels have a truck parking area somewhere behind the building. Not always convenient, but there.
Moab: That time of year it should be no problem to park in the City Market (brand name; not location) lot and walk to town (can't really call it DOWNtown.)
Torey: You would probably be ahead to use a private CG. One on the East end of town or the one at Sand Creek. (Nice people were running Sand Creek the last time we were there.)
Final thought: We were there in early March about three years ago and had to get the fiver out to Green River in a driving snowstorm. Once we got over the mountain to Grand Junction, the weather was cold but otherwise fine.