Firstly, I'm so sorry he's put you in this position, knowing what your little family unit means to you, and how desperately you'd never want to put your children through what you went through. That's one of the most heartbreaking parts of this whole sh*t show, and one I'd personally find hardest to forgive.
Very sadly, his message reads as more self indulgent twaddle. The first half is designed to tug at your heart strings and soften you up. The second half is him wittering on about how much it'll hurt him to leave the OW.
If you want to stay in the marriage, then I agree with PPs who say the actions now have to come from him. Not in response to an ultimatum from you. That will be the answers you're looking for - in his actions .
If you want to know who somebody is, watch what they do, don't listen to what they say.
If you stay, you need to build the trust back up somehow, and how can you possibly start to do that if he doesn't willingly choose to do everything he possibly can to show you how much you mean to him?
If he doesn't, you'll always be wondering what he really wanted, what he really felt or feels, just like you are now. And that, presumably, is why you want to know if he's messaging with the OW? Because you're trying to work out where you stand. You'll never know unless he makes it very clear by showing you. And he hasn't shown that yet.. Except that he still hopes he can keep her friendship, and you and your girls.
He needs to walk across hot coals for you and your girls now!
And you may well find that however much you're longing for what you had, that's because you're mourning what no longer exists. And there might be nothing he can do to repair this.
It'll come down to what you want, which is complicated, multifaceted, and will keep changing as you transform over the course of all this. Until the dust settles..