The girl I am seeing sometimes goes to 3-4 AA meetings a week. Do people date each other a lot at AA meetings or am I just being overly suspicious?
It is recommended that newcomers to AA not date at all, in or out of the fellowship. If she does not have a year or so of experience in sobriety, the distraction of a relationship could cause her to relapse. Be glad that she is doing what she needs to do for herself.
Your relationship is a different matter. Generally speaking, if you do not trust someone it is either an indication that they have done something to lose trust, or that you are dealing with a personal issue yourself. Only you can decide about that.
Why not go to a few open AA meetings with her to find out what her AA life is like? It might ease your concerns, and it would assure her that you want to be an active part of her recovery.