Adult relationships can be just as complicated as teenage ones, because many folks have emotional scars from previous relationships and complicated familial relationships – especially if they have kids.
Dating in my 30s and 40s was no picnic, as I am introverted, socially awkward, and had an abusive childhood making those things even worse. After my divorce I was terribly insecure and had a lot of deep issues. I was a nightmare to date. Eventually, I recognized my behavior was self-sabotaging and also harming my child’s emotional development, so I stopped worrying about being in a relationship and focused on myself and my son.
When I felt stable enough to date again, I found that I was better able to detect when the guys I was talking to had issues of insecurities or ego, and I would not pursue a relationship with them. Eventually, I gave up looking, because the majority of people ‘seeking’ relationships seemed needy, and neediness is bad for relationships. It was then that I encountered a gentleman who shared similar interests and we struck up a friendship and realized that we have great chemistry and compatibility and we decided to build a romantic partnership (we just passed the four year anniversary of our first “official” date, we are in the process of buying our home which we’ve shared for three years, and we’ve been engaged for almost 2 years. College financial aid rules for my son’s education are the reason we aren’t married yet).
The key is to not act like a teenager, and wonder what she’s thinking; but to act like an adult and to ask her what she’s thinking. Mature adults communicate and take accountability for their own actions, and they do not tolerate games or immature behavior from others.
In the situation you describe, I would straight out say, “I really enjoy your company and friendship, and would like to get to know you better. However, I am getting mixed-signals from you, and I’d appreciate if you could give me an explanation and let me know if you want to be friends or if you’d rather I leave you alone.” Place the ball in her court and step back. If she does nothing after a few days, move on. If she comes back sincerely apologetic and expressing reluctance because of past experiences, then it is up to you if she seems reasonable enough to be able to overcome those worries, or if she seems too irrational and immature at this point in time (such as the ‘damsel-in-distress looking for her knight-in-shining-armor).
If you decide to move forward, I recommend keeping the relationship platonic until you are certain that she is stable enough for a healthy romance and that you are both compatible in expectations for how a partnership works (finances, chores, all the little things that make a big difference in the day-to-day of real life). If you lead her on and allow romance to enter the situation before you you are ready to make some degree of commitment, there is a good chance that one or both of you will wind up hurt. That doesn’t necessarily mean waiting months, because again, you will be assessing her stability and your own level of interest as you move forward. But as long as you are uncertain about anyone’s feelings, it is best to stay platonic.