It sounds like their every other weekend policy isn't really the policy for everyone, so it would seem you have a chance to get your Sundays off. Also, if that was an agreement during hire I would think you could bring it up to your boss.
I've never seen a place that was flexible about the weekends. In all the hospitals and other facilities I've worked at, everyone except management rotated every other weekend, and where I'm at now, you can't even request one of your weekend days off. You have to find (non existent) coverage yourself. It sucks because, basically, if anything comes up that you'd like to go to on your weekend, a concert, a child's recital, a sporting event, whatever, you're pretty much screwed. :/
But as far as church goes, until it gets sorted out, you could go to church on your Sunday off, and then on the Sunday's you work just watch church online with the family Sunday night, vs. being stressed about being late for work and dealing with potentially irate coworkers waiting for you, etc...Good luck.