If he is masturbating a lot, unfortunately that doesn't mean there isn't another woman. Quite the contrary. You'd think if he was just so stressed with three small kids, work, and a failing marriage, that his drive would be dimmed. But it's pretty common if someone starts a new exciting affair that they will find their drive increase a lot, and end up acting on it a lot. Doesn't mean he isn't meeting another woman, or doing phone stuff, or whatever.
Regardless, as others have said, if one person says a relationship is over then it's over. He's done, and been pretty clear he feels for you more as family than a romantic partner. It happens to people, especially with such overloaded plates like three tiny kids with a very short gap between them.
I would tell him you're disappointed but understand and respect his decision, so let's sort out practicalities. Obv you'll need a few months to job hunt and find something, and for you both to think about housing. And to figure out custody, and a 50/50 split. Just crack on. Pining after him won't make him want you more, sounds like it's done as far as he's concerned. I would be grateful you have your beautiful children (it's much easier splitting with young kids imo than splitting without any kids and worrying you've missed your chance to have them! You have them for life, and if you want to date again it won't be with the pressure of tryna find a potential father for kids, if that makes sense).
Not to suggest games, but doing a 180 and becoming very nonchalant about it all and acting unbothered and a bit disappointed and like you'll be perfectly great without him is pretty much the only way that he might decide to change his mind... begging and desperation is offputting to people. If he wants you back you can decide how you feel at that time, but for now? Be a strong, confident, woman and mother. You don't need him.