How do I deal with this new girl?

Hey… so I met this girl at a party who I think is stunning. The night I met her I talked with her for over an hour just about random stuff. A week later I asked her out on a date… we went for dinner. A couple nights later she was at a bar with some of her friends and I was there too and she sought me out (we knew ahead of time we’d both be there). We had a good amount to drink (nothing crazy) and she invited me back to her place to just cuddle and kiss a bit. We went out a couple more times and the same kind of intimacy has been happening where she seems really into just being close to me without taking things too fast.

The reason I’m not sure about where this is heading is kind of complicated. I’ve never been in a long term relationship and have had issues the past 4–5 years with mild depression. I’ve had trouble meeting people after having some abandonment issues in my past with good friends, a rough parents divorce, etc… I’m trying my best to show her I am a good guy when I’m around (paying for her, walking her back to her place, comforting her) but I just don’t know what she’s thinking about me. Obviously I have a lot of insecurities about myself and getting close to someone that I feel sometimes it shows in how I interact and this leads me to believe she finds something off putting about me. I fear she may not want to see me again or may lose interest if she already hasn’t. She also seems to be a bit more adventurous than me (I’m a more serious, introverted, isolated kind of guy).

Any general advice would be great. Like I said, I don’t know her too well, but it seems like there is a bit of a spark. I don’t know if it’s just because I find her very physically attractive or if it’s something more.

One thing I’m trying my hardest not to do right now is appear clingy or needy. My instinct is to message her all of the time to check up on her. I’ve restrained and I don’t think I’ve so far shown any signs of being clingy but I sure want to be.

Let me know what you think I should do and how I should maybe find a better mindset going forward because I fear she’ll leave me once she gets to know me a bit better and some of my issues.

Thanks guy.