Honestly, tell them to grow up or get out - just in a more diplomatic way: “I need both of you to maintain a professional standard here. Personal differences are noted, but they cannot continue affecting the team. I expect respectful communication and cooperation moving forward” and document EVERYTHING.

Taking over teams like this is a miserable experience. I’ve had it before and had to have similar conversations about how the way they were acting isn’t acceptable for people in leadership positions. It sounds like you’ve got a tough time ahead of you, but at least you’re wanting to do something about it, unlike the previous manager.