L.R.
A., you MUST ask these questions of your attorney -- get one if you don't have one.
Getting married means bringing a new and different adult male into your children's lives. If you have issues with your ex, he absolutely could make your boyfriend/fiance/husband an issue, especially if you are not open and above board about the existence of this man who will be living with your children at some points.
Please get an attorney's answer. Don't depend on us, here on an online forum. We are not lawyers and you need to be certain of your legal rights and your responsibilities too. You also need to think out with your attorney whether your ex can or would use your engagement or marriage against you in any way. And the financial impact of getting married is something you surely must report, just as you have to tell the IRS and the state.