My (23/M) partner (19/FtM) has started talking with their ex again and I don’t know what to do

I’ve never been in a situation like this so I could use some help, what should I do here? Basically my (23M) partner (19FtM) has started getting on friendly talking terms with their ex and it makes me feel really uncomfortable. For context, they were together for 2 or so years on and off before officially ending things last January. I thought it was clear and mutual though not long into our relationship, I dropped them off at their ex’s house to talk and they ended up wanting to be picked up early in tears. They felt bad about how they ended things with him and that it wasn’t as mutual between them as I thought. Fast forward a bit, and we’ve been together for 3 months now and I feel so committed to and in love with them. Only issue is that they started talking with their ex again over DMs, seemingly in an effort to get on better terms, though the fact that they appear so desperate to be amicable with him rubs me the wrong way. I’ve seen out of the corner of my eye how he’ll send them cute animal videos and stuff like that, as well as how they have the heart background on their DMs (Instagram feature), which they said was just leftover from when they were together and haven’t got around to changing it. Lastly, I’m going to be moving about an hour away from them soon, and I just don’t know how to deal with this, especially given the fact that I won’t be seeing them as much. They tell me they love and care about me all the time, and I believe them, though I can’t shake the feeling that I’m just in a rebound relationship until they fully warm back up to their ex. I keep having the thought of proposing an ultimatum, asking them to stop talking to him or I leave, but that feels excessive, and I really do care about them and don’t want to lose them over something like this. What should I do?

TLDR: My partner has started talking to their ex again after ending things in a not-so-mutual way, and now I can’t get rid of this paranoid feeling that they both still want each other.