If you're on a month long overstay, I wouldn't go to immigration. If you're caught in country on overstay you can be arrested and deported. Whatever you do, don't go over the 90 day mark.
You have to be really careful man, if you overstay more than 90 days you'll get blacklisted for 1 year. Month is a long time. I wouldn't go to immigration yourself, I'd let an agent contact them for you.
Also, once your child is born you'd be eligible for a Non-O based on having Thai family. You have to make it here that long though and don't get blacklisted.
Depends on how much of an overstay. Some offices will let you pay on overstay fine and get an extension (especially if you've tested positive for Covid and have been isolating).