The shortage of refs isn't a quick fix.
I think the assignors and teams need to try to coordinate certain days for certain classifications and give them the priority. A and AA on Mondays and Wednesdays and AAA-AAAAA on Tuesday, Thursday and Friday, or something like that. A region game should take precedent over a non region game.
Not saying it will work but it's an idea.

Fall soccer won't work for high school because they you are asking club soccer to play in the spring. Too many moving parts and too many egos to make it work.