Can I just point out that if court is on April 1, Mom was served long enough ago that this is not NEW information for her. And she has a specific period of time in which to answer the motion. My money is on the fact that the motion was NOT answered... which already puts mom at a disadvantage. Add that to the fact that she DID move w/out notification, and she DID stop taking the kids to visits, and it gets worse.
Mom, what you NEED to do is
1) find an attorney to help you. You are not capable of doing this on your own. Not because you're stupid, but because you lack the communication skills required to get your point across coherently.
2) concentrate on whether or not there has been a change in circumstances since the time of the last order. Not even a change w/dad... a change w/YOU or the kids. We know you moved. How far from where you were? Did it make Dad's ordered visitation easier or more difficult? How are the kids doing in school? Are you providing for them adequately? Are you remarried/living with someone?
3) I'm going to assume Dad's conviction was prior to the last order, and that's why the visits are supervised. Make sure the judge is aware of that situation.
4) Understand what he's asking for. If all he wants is joint-LEGAL custody, that's just joint decision-making about things like major health care, school, etc. Unsupervised visits are unlikely if he's not exercising his supervised visits... UNLESS he can prove he didn't exercise them because of YOU.
5) FOCUS. No one here is bashing you. But YOU have to understand that YOU will be held accountable for your actions. And HE will be held accountable for HIS. And the JUDGE is bound by FACT and LAW, not WISHES.