Anxiety after first fight in new relationship

I (28F) have been seeing this guy (27M) for 3 months and things have been wonderful. We’ve both shared that this is serious for us and we see this becoming a long-term commitment. Last weekend (before Thanksgiving) we went on a date and were drinking; I was drunk and took a joke a bit too far and it was insensitive and hurt his feelings. I apologized right away, but we were both drunk and it just wasn’t a great situation. We always spend the night together after dates, but he needed some space and asked me to go home by myself afterwards. I didn’t hear from him at all the next day (Sunday) which was hard for me as someone who is anxious and likes to talk things out right away; I FaceTimed him and he didn’t answer, so I texted him on Monday morning and just told him that I was checking in and was so sorry for my behavior. He answered and we ended up talking it through over text (which is not preferred on my end; I will always prefer talking in person but I was just glad to be talking about it period). I was still feeling a bit anxious the next day, so I asked him to FaceTime and he confirmed that we’re all good and seemed totally fine.

We have not been able to see each other since the night the fight happened because of Thanksgiving, and have only FaceTimed/texted since then and my anxiety has not gotten any better; now I’m constantly reading into everything and wondering if he still feels serious about us or if I’ve ruined things. I hate this anxiety and my question is, how do I move past this? I’m seeing him tonight for the first time in person since it happened; I feel like I need the extra reassurance in person, but I also don’t want to push him away and don’t want to insult him by seeming like I’m not taking him for his word that we really are ok, especially now that it’s been over a week since this happened.

TL;DR: I’ve (28F) been seeing a guy (27M) for 3 months and we had our first fight, we talked it through and although he assured me already that we’re good, I’m still struggling with anxiety afterwards and don’t know how to move on, and am hesitant to ask for more reassurance because I don’t want to push him away or insult him.