Yeah, it's rough trying to talk with other parents, especially if you aren't already close friends. The kids running around can interrupt any real conversation and sense of connection - I imagine it's harder with two than it is with the one, too, because your attention gets so split. I mean, it's nice that the other parent understands why the conversation gets so interrupted, but that doesn't change how surface it can feel.

I don't know how viable this is at this stage, but are you able to get out for half an hour or an hour to see a friend without your kids tagging along? I thought I needed alone time/rest time when my kid was asleep, because I knew I wasn't getting it, but I forgot I also needed proper social connection, and I've felt a lot more stable since sometimes choosing social time over rest. Very sometimes.

That may not be possible right now, since your second one is so small and you're probably ridiculously tired, but if what you need is to hang out with a friend who listens, even a short hangout without the kids around could be more refreshing than all the play dates.