Welcome to the life of a freelancer! Suggestion is to become an expert in juggling.
On a serious note, though: is this a season of shows at the same theatre or with the same producing organization? You mentioned the director is the same for all shows, are the production teams also the same? If it's the same overall team, you should have an easier go at it as you'll all be in the same situation and the producer and production manager will collectively aid in finding a way to make it work.
If you are juggling between three different producing companies, production teams, etc, then you are in a bit of a different boat as you will have to find a way to please all comers and be a better juggler, as saying "I'm swamped with Fantasticks at the moment" won't fly if the producing organization for the other shows needs your attention. In this situation, staying organized and managing potential scheduling conflicts as far out as possible will be instrumental to success.