What you have described certainly fits the bill for some sort of GID, or perhaps just plain ole BID. You certainlyy are not unique in having those feelings and the mass confusion that comes along with them. I've been wrestling this demon for some 50 years. Someday I hope to have an answer that works for me.
Much of what you have described fits me, as well as many others here. I felt when I was 4-5 that I should be a girl. Prayed each night for a miracle. Failing that meant doing my best to fake being "Normal" which meant a very rigid and rule bound life.The shame and guilt led to pretty much believing my entire life was a fraud, a sham, a lie. Including all the pretty amazing acomplishments people I respect tell me I had. Yet, they mean nothing to me since it is all built on a lie.
You did a great thing in actually bringing this up to your g/f. You are only as sick as your secrets they say. Very true for us, as you noticed the sense of relief after telling her your deepest darkest secret. Unfortunately, with your confusion over what you are with many years of wrestling with that question on your own, you cannot give good answers to any of your g/f's questions.
Job #1 is now to sort these feelings out. THere is a broad spectrum from "Normal" to true TS. (There are several classification systems out there) Also keep in mind that if the day comes you are sure you are a TS, that still does not mean a full transition and SRS is required for peace of mind. Again, plenty of options.
One of the best ways to find a gender therapist, if you are not near a major metropolitan area, is first look for TG groups nearby. You will likely find that they can be of infinite value. All that shared experience and, at the very least, you'll learn which therapist to stay far away from.
If you are near any sort of metro area, odds are there is an organization which specifically is for the support of all things trans such as therapist, medical, legal, etc..
You can try posting in the Therapy forum for leads. As for finding a TG group I spent about 3 months of evenings on Google searching for one in my area. The closest iturned out to be 90 miles away. Same for any therapist that states treats TG clients. That is what I get for moving to the hicks from 10 minutes away from mid-town NYC

It's about 2 1/2 to 3 hour drive for me to get to a big city gender clinic with all the whistles and bells.