Co-parenting with BPD ex for 3 yrs after div- until I announced my engagement

[SWPoet]

I have an almost 16 yo son and 12 yo son. Their dad and I have been relatively cordial for the past three years and shared expenses without sending receipts, just said I paid this part, will you pay this one.  No drama. Or not much.  I went 11 mos not dating anyone and then met someone who also left a BPD longer term marriage than mine.  We are getting married in the spring. My kids like him and he is a good person, but my oldest thinks when he turns 16 in a few weeks, he can just go live with dad (we have joint custody) and now their dad is also sending me emails with what he expects me to pay (half of) without considering all that he hasn't paid.  Anyway, I've tried the gray rock thing (with narcissistic personalities) and he has now commented on how I don't want to answer his calls.  I work where I take crisis calls (social worker) and can't always answer but he also blew up at me a month ago in front of the kids (never did it that bad during 15 yrs marriage) for very minor issue so I really didn't want to discuss things with him where I didn't have proof of the conversation.  

We split mutually, without court, and with joint custody (legal and physical) and no exchange of child support or alimony.  It has been spiraling since three days after I told him I was engaged (to which he calmly congratulated me  and then found something to blow up about three days later).  

My question is... .
How do others deal with the narcissistic/BPD ex and co-parenting when you have recently announced engagement or have remarried.  Its been three years and he was more the "silent treatment, gaslighting, type spouse.  Also, how do you deal with teenage boys who think their dad does no wrong (because he never admits any faults)?

Any suggestions would be greatly appreciated.  

SWPoet

[Turkish]

Good that you filed a stipulation without going to court.  I was able to do that as well,  joint legal and physical custody.  What does the stipulation say about splitting costs (medical, activities, etc.), if anything?

How did your sons respond when he went off on you like that?