“The Gay Deceivers” is about two heterosexuals who pretend to be gay to avoid the draft. It’s a fairly homophobic film and does not present gay people in a positive light (despite the best efforts of co-star Michael Greer to get rid of some of the worst of it) and doesn’t touch on gay relationships at all that I remember. “Boys in the Band” has one co-habitating couple in it and explores their relationship issues in about four minutes. While both films are IMHO worth watching they aren’t going to be what the OP is looking for.
A film from that era that deals with an actual couple is “Staircase” with Richard Burton and Rex Harrison (and Harrison’s character is up on charges for soliciting a police officer in a public toilet so it’s even topical) but I can’t recommend that either. Or for the ladies there’s “The Killing of Sister George” with Beryl Reid as an abusive bulldyke who loses Sue Lyon to the predatory Coral Brown, or “The Fox,” based on the D.H. Lawrence work in which Sandy Dennis takes a tree between the legs so her girlfriend can run off with Keir Dullea. The late 60’s/early 70s weren’t a great time for films about stable same-sex couples. Of course the 80s aren’t much better. Off the top of my head the only stable same-sex couple I can think of is Michael Caine and Christopher Reeve in “Deathtrap.” Then we get into the 90s and New Queer Cinema, which wasn’t really all that interested in exploring monogamous gay relationships as a theme.
Oh, another possibility is The Trip from 2002. It follows a couple over the course of about 25 years, from the 70s to present day. It is rated R for language, nudity and drug use so may not be the best choice for a church group but there it is.