The common push-pull trap

I'm sure this is brought up extensively, but I'll keep my situation short.

Essentially I'm the one in my relationship to close the gap and connect in the moment that something happens (potential upset/hurt), but my partner shuts off in the moment. I'll ask them what's wrong and they lie and say "nothing, it's fine," but then reveal to me later what they were upset about.

They're now expressing concerns of feeling disconnected/lack of emotional connection. They're saying my sensitivity interferes when they bring up an issue and worry too much about it upsetting me or hurting me and they hate to see me that way. However, I've told them that my sensitivity isn't a cause to stop the conversation. I'm actively listening to them and trying to hear them out.

So ultimately they want more emotional connection, but when I try and connect in the moment, they get avoidant. Even now, they want to resort to only texting me instead of having a conversation in person. Yet they say that they're not sure why it's so hard for me to have a serious conversation with them.

Have any of you experienced this dynamic before? I now know what the focus of our current issue is and I think I need to curb my emotional response and they need to trust that they can be open to me, regardless of their feelings potentially causing hurt.