In my league, I have conference movement off. I went to move a team to my conference using the realignment tool on May 1st. It wouldn't add the team I wanted to my conference and now that I'm thinking about it, does having "Conference movement" off make using the realignment tool not work?

No it does not - what do you mean it would not add the team? Did it give you an error message? Did you first change the conference setup so that it would be one team larger?