Relax and focus on getting to know her as a person before you decide how serious it can get. It sounds like you two may need to slow down and just enjoy each other's company.

I had a kind of similar situation a few years ago after my boyfriend died. I met a guy a month after he died and I was looking for a friend not a boyfriend but I kissed him and things went too fast after that. I had only known him a few weeks when he started telling me he loved me and I told him he didn't even know me yet but he kept on being clingy and I stopped answering his calls.

There is no way of knowing yet how this person may fit into your life (or not) but be honest and try to be friends first and if something else grows great but maybe it won't.

Good luck. It sounds like you are on the right track and have good self awareness.