Dating isn't the answer to your self-esteem issues, you need to fix that from within.
Which is a lovely glib little remark innit OP, I don;t mean to annoy or dismiss you with it, & appreciate it's easier said than done.
Firstly I agree with Morewine about knocking his visits to the family home on the head. It's not good for you to be playing happy families in your own space. He forfeited the right to have you facilitate him when he betrayed you, just stop doing it. he can take the DC to his flat or entertain them at various venues.
Secondly - don't lie to DD12 if she asks about the affair again.
She clearly suspects, & has suffered 1 parent effectively leaving her - you need to be her safe space, her font of truth & no bullshit, who she can rely on not to let her down. Lying for daddy is not helping. any more than allowing daddy to waltz into your home every night is.
You don't have to give her the unvarnished truth, but you can easily say something like "he made some mistakes & lied about them to me, but you'd need to talk to him about that. Also it's me he's left - not you, & whatever has happened between us, we both still love you" etc.
Thirdly - you are only 7 months in.
It's going to hurt.
It's hurting worse, & longer, because fuckwit is inserting himself into your evening routine every night. You need clear air between him & you, or how else can you start healing?
Finally - and I’m trying so hard to keep things amicable
Fuck that noise.
It's not amicable.
It can be civil, in front of the kids, & about each other, but there is no amity here. The cunt bought HIMSELF an STI kit. Presumably the possiblity of you being infected didn't bother him, so long as HE was ok. He coldly planned whether to leave before or after your 40th. He lied, plotted, spent family money on his floozy. He broke his vows & his commitment to his DC, all to get his leg over somebody else.
You need CHUMP LADY OP.
https://www.chumplady.com/about-chump-lady/
Have a browse of the archives. Marvel at the wisdom, experience, helpful advice & snark.
Find your anger.
Reclaim your home space. He no longer gets to enter your home.
Date your Hinge guy if it makes you happy, but don't lean on him - he;s your rebound. Have carefree sex with him & don't look to him to be your sole source of validation.
Plan a holiday with the kids. Focus on your career. Make sure you see friends & have a laugh. Get a divorce pushed through, do whatever you need to do re: settlement, whether family home needs selling etc.
All of this is about taking steps toward a fresh start, which you won't get while you let your cheat hang round YOUR home, disturbing your peace of mind, playing at amicable.
Fucker has no amity for you. Don't believe a word he might say about financial settlements either - he's a known liar. Lawyer up. Get angry. Get what's rightfully yours. Get free. 