Mandatory OT for PRN?

Hello! I am in quite the pickle & looking for advice on how to handle my work situation.

I am a newer nurse and stayed with my 1st job for almost 4 years. Due to new family obligations I had to make the switch to a PRN position in a different hospital. I'm available to work 24 hours / week, occasionally 3 and pretty flexible as well.

Since being hired on the unit has become short staffed & I have been scheduled not only FT hours but OT as well. Just random and continually changing surprise schedules with 5, 6 and even 7 (12 hour shifts) in a week.

I have had to stretch my support system to the max to keep this job - it's not sustainable for much longer. I have spoken to management - they acknowledge they understand, but the schedule is what it is and it's on me to find coverage. PRN status doesn't matter at all. I put my foot down and said ok, I'll work with you guys as much as I can, but absolutely no overtime. I cannot sustain this with my obligations outside of Work. To my surprise I've been scheduled OT again.

Perhaps I am missing something? Has anyone heard of PRN's in my situation? I'm all for paying my dues and I'm a hard worker but I also have obligations outside of work that are non-negotiable. I actually love the job as well and I'm a good fit besides the hours.

If I quit my new job I burn my bridges with this hospital ( rural area - limited opportunities) and there's no guarantee I can find another job quickly. I also need a good reference so having a potential new employer call my current would be a dead give away I'm going to put my notice in. Any advice on how others have handled a similar situation?