You are in your 30s and employed full time. That's a better place than I was in when I left my husband and I can tell you that kids or no kids, you can and will survive if leaving is what you choose. So far you have reached out to his friends and relatives. Where are your friends and relatives?
You can send your kids to college. If you don't have the money to do it, they can get scholarships, they can work part-time. If you are afraid that they might turn to drugs or alcohol, well, having exposure to an addict parent might make this outcome more likely than if you leave your addicted spouse (which seems to be what you want to do, but are afraid of doing). If you are afraid of his new wife preventing him from seeing the kids, maybe that's for the best... but I really think you're thinking too far ahead here as none of this has even happened.
You have a house you can sell -- that's an asset. In 10 years when the spousal support runs out, your kids will be teens and it's only 5 years until the youngest can legally leave home.
Just think about your situation right now: your husband is back with you and struggling with his health... you feel resentment towards him and want to leave but are afraid of leaving. Do you think (be realistic here) that your husband will recover and never relapse... ever... again? Do you think he will be faithful from now on? Or has he shown you who he is? How long can he be "good"? What happens if you stay with him for another 10 years and then he relapses and also cheats again? By then the kids would have had 10 years of walking on eggshells around him. If you're 35 now, you are still young enough that starting over with someone else is a viable option (after you have had time to recover from this relationship... that is, if you actually want to start over with another person... you may just want to be single).
I had to start over in my 40s. If I could go back in time, I would have left at 35.
Of course, you have to do what you feel is best for you and your kids.