Hi Maggie. I'm sorry for all the grief you have to deal with. I don't see how you and the two kids who are in your home can not be angry. He turned his back on his responsibilities. Not much of a man. You had ample reason to end the marriage, when he quit his job. He's gone home to his folks to begin a second childhood.
You say he still drinks and lies. Isn't that why you divorced him. I'm confused about why you would think the drinking and lying wouldn't continue.
Your 2nd issue is his being in a new relationship. What kind of woman would want him? I doubt those two are going to build a great life together.
The kids have something to be rightfully resentful of. The upside for you and the 2 kids is that you don't have someone staggering home drunk all the time. You were on a roller coaster that you wanted off of. Those 2 kids were on that same roller coaster. The home should be a nicer place now. I lived with a problem drinker for 7 years. Then I left and was much happier alone.
You have the heavy burden of raising you kids with little help from him. That is a tough challenge. You'll do the best you can, and the kids will know who really loves them.
You all need time to adapt. Life with him gone may prove happier.