I don't see where you are stuck.  As an organization you have the choice of choosing whom you do shows with.  Just don't do shows with that other group.  You can still have a working relationship without commingling for shows.  Put in your advertisements a section stating no parts have been pre cast all parts open.

I feel your pain. It's still a question mark who/how many people we'll get at auditions, and scrambling to find people is not unusual, (unless it's a big name musical show). My suggestion is start small, if you know you won't get big turnouts look to do shows that are well known but don't require a big cast. Then make a point of using the people you get. May not be the cream of the crop to start with, but you could also get those diamonds in the rough that the other theatre hasn't tapped into. Over time, hopefully, people will realize you are putting on quality shows and those actors who are used to being handpicked will get interested and show up for an audition. Just focus on the quality of productions and making it the best experience for those who do audition and get cast.

I think one of the questions you have to ask is why aren't actors auditioning? Do they feel that auditioning is beneath them? Are they intimidated by the idea of auditioning because they are out of practice or inexperienced with auditions? Could it be that they are reluctant to initiate participation with so many undetermined variables i.e Will I be cast? What part will I play if I am cast? Who else is in the show? etc.

If you can determine the influencing factors then you can develop a strategy to speak to them.

I was thinking it might help to appeal to them through the work. Choose plays that are known to draw the kinds of actors you're seeking to recruit- you know, those juicy or fun roles that everyone can't wait to get their hands on. Find shows that make them think, "I gotta be in this show!"

I really like chelserin's idea of sending out personal invitations. It's a nice touch, and may go far to soothe egos as well as fears (if either one should be a factor).

I don't know much about your community so I don't know what it would take to get their attention, but just off the cuff, and this may be reaching a little too far.. but what if you tried to make audition more "social gathering" in atmosphere with some food and drink (this could work well combined with the invitations). Since it sounds like these are more seasoned performers- instead of cold reading maybe prepared auditions would work well for you (or a combo of both).. and create a more performance, dare I say, talent show atmosphere for the audition.   

Sometimes if you can get just a few key peoples buy in the rest will follow, and creating those personal connections can go a long way.