I'm not going to cover shifts
Why don't you want to cover shifts?
Because I deserve to be a leader there and am not getting what I want
I understand your frustration with your career.  However, you're not going to get anywhere without being a team player.  You should be supportive of your colleagues instead of undermining them.
I'm not undermining them. I just don't see the point of continuing to go the extra mile for this job by covering all their shifts when my boss doesn't recognize my hard work and won't promote me.
I understand, and it sounds like it might be a good time to change your focus, or your job. After all, we need to take care of ourselves. Your time off is important, too.