I'm alarmed that my ex-wife got engaged to an unknown guy that hasn't met our three kids we have joint custody of
I am sorry to hear you are going through this. Are you afraid of losing her and your kids to him?
I can't imagine her with anyone else than me
While it's fine to worry about your kids, whoever your ex-wife dates is none of your business. I'm sure that she would not let anything that would harm your kids, but if you're still worried about them you can try and communicate that with her and set some boundaries about the interaction with the guy that your ex is dating and possibly limit his interactions with them.
I try to be nice and willing to get along but she is acting like a selfish child.
It is good to work on finding acceptance for what your wife is doing. You really only have control over your own actions at the end of the day.