You sound a bit too nice!!! It's a good thing for your ex, but what about for your kids? He helped created them too so why should they suffer because he didn't want to pay, ya know what I mean. You need to go to your local Office of Support Enforcement (or whatever it's called where you are) and have them help you. They will set up an amount based upon both of your incomes. They track the payments for you. This way he won't be able to come back on you and say lies. I don't know him to say they he would but people do strang things when it comes to money. This money is for your kids to live on. Sure, you can accept his offer for now, Verbally only. Do not do anything in writing unless you have legal aid. Don't be a sucker.
You've gone this long without his money. What's it gonna hurt to do it the legal way. This doesn't mean you need to go get a lawyer, yet. Go thru Legal Aid or thru the local child support office.
Good luck! Hope you can work it out fairly for the kids sake.