Hi metalchick. Sorry for your situation. With two kids to be cared for, this isn't going to be easy. Things may not be as bleak as you fear, though. I think you're making some unwarranted assumptions.
First of all, a father of two children doesn't get to "wash his hands of any financial responsibility." He will be liable to pay child support. He will fork it over, or the state will take it right out of his check and send it to you. The courts are very interested in the welfare of children . . . much more than they care about his supposed claims to "community assets." So he might want to be more compliant to your wishes. If you are interested in keeping the kids with you, let him go right ahead and demonstrate a lack of commitment to being an equal partner in parenting responsibilities. The court will take that very much under consideration.
I'm no legal expert, but, if you think you can afford to maintain this house as a home for those two children, I believe the court will be supportive of that. Common law marriage status us not as automatic as many believe. Here's a link that discusses the way it works in Rhode Island:
Common Law Marriage in Rhode Island | Fact or Fiction. Look through your paperwork to find any validation you can that helps establish who contributed what to buying and maintaining this house. Make a list of what each of your incomes were over those 16 years. Did you write checks to pay the mortgage? Did that checking account hold only your wages, or both of your wages? What did he do with his earnings over those 16 years? Was he prone to buying himself big-boy toys? Make a list. A lot of the past can be resurrected, if you think hard and come up with evidence of who did what with their money. Was he unemployed for intervals? Write out a little history of his and your employment over the course of you two being involved. Did his or your parents contribute to household finances? Judges don't like to be played for fools by some jerk who runs out on a family and thinks he's all entitled to whatever. I think you'll find the Law makes a lot more sense than that.
Is your ex's name on the title to your home? Since a piece of real estate and the welfare of two kids are at stake here, I would think an attorney's counselis essential. But educate yourself as much as you can. Start googling stuff. When you visit a lawyer, be well organized because the more of the lawyer's time you take up, the more you're probably going to pay. Have your questions written down. Be prepared to take some notes about what the lawyer tells you. I have no great advice on picking out an attorney. The few times I went to lawyers, I usually didn't think much of them.
Here's a thing to remember about lawyers. They don't make any money from people getting along nicely. The more you two can be reasonable with each other, the less you'll both end up paying in legal fees. You might pass that little nugget on to your ex.
Keep a little journal of what goes on between the two of you from here on out. Write down when he spends time with the kids. Make a note of any bad behavior he exhibits. Keep an account of what money he contributes to the expenses of raising your children. Our memories fail us when we're under stress. Little details that might seem insignificant to you in the moment, might be of greater import when you are negotiating custody and child support.
In matters that end up in court, things tend to go best for those who show up most well-prepared. Your ex sounds like a guy who never put much preparation into anything. Let that work to your advantage now. And don't go feeling sorry for this guy to your own detriment. If he finds his threats don't control you, he'll likely work some angle to get you feeling like he's the victim.
I hope things work out for you and, most importantly, for the welfare of your two children. I think you'll find, that's what the courts care most about too.
One last thing I thought of. Nobody knows lawyers like other lawyers. See if your parents have ever used a lawyer for anything and had a good experience. If so, see if they can ask that lawyer to recommend someone skilled in family law.