I feel I'm too clingy. Should I talk to her about it? Or should I just let things go for now? Better call my therapist...
You don't need to call a therapist silly.
Honesty is very required in a friendship or even in a relationship, it does make things better to be honest if the two of you can handle a -real- meaningful relationship. Thing is, if you do feel clingy then you're probably being clingy. She might not mind at first but eventually she'll feel suffocated.
Give yourself some space. Without making her feel like you're being distant. It's tricky, but just take things slowly and cool it off, don't obsess otherwise you'll start coming off as controlling.
Also no need to go all "Imma call mah therapist". Relax, everyone has insecurities. Talk it over with a friend.