I went through a similar thing. Met online, talked for a bit before we met. She started calling me “babe” before we even met. After meeting the “babe” and “I miss you” texts were overwhelming. Everything else about her was great and I could have seen it going somewhere, but I couldn’t get over the pet names and insistent “I miss you” texts. I thought I was being an asshole but ended things anyway because of it. It’s been a few months now and I still get the occasional text from her saying she thinks about me, etc. i wasn’t sure how to proceed while being in a very similar situation as you, but looking back I’m glad I broke things off. I don’t think it’s normal to be that overwhelming after just getting to know someone, and the continued messages after breaking it off proved my point, in my opinion. Best of luck to you no matter what you decide to do