Please give yourself credit for coming through a difficult childhood as well as you did. Despite the lack of support, you managed to get a good education and a job and you’ve been in serious relationships. Not everyone from such a dysfunctional background has been able to achieve so much.
It’s not at all surprising that you find it difficult to lead what you believe to be a “normal” life. Children learn what they experience. A child from an emotionally chaotic background generally doesn’t learn the coping skills most of us need to handle the usual challenges of life. Further, it is difficult to learn how to cooperate with others, how to respond to normal emotional needs and how to communicate with others when the models for doing so aren’t able to do it themselves. That probably has a lot to do with your lack of friends or a significant other.
I do think you should take it slow with your girlfriend. “Liking” isn’t enough of a reason to marry. Your statement that you will be okay with it if she breaks up with you makes me think that the relationship isn’t deep enough or developed enough to be considering marriage.
My guess is that your girlfriend can understand why you might want to “hide” your family background. But neither of you will be able to forgive you if you don’t do what you need to do to recover from it.
From what you wrote, I don’t think you have a mental illness. But you do have a serious lack of people skills as a result of your upbringing. For that reason, I recommend that you work with a therapist for awhile. With the support of a therapist, you can learn and practice the many skills you weren’t taught as a child. Having such skills will make it far more likely that you can be successful in marriage as well as with friends and on the job.
I wish you well.