I don't think you're over reacting at all. Things happen sometimes and you can't take them back. But in my opinion it's not the mistakes you make in life that define you, it's how you respond to them.

Giving your girlfriend the benefit of the doubt that this was just one of those stupid things that you do when you've had too much to drink the fact remains that this hurt your feelings. When you care for someone, you feel bad when you hurt their feelings even unintentionally.

I'd examine the frequent fights you've been having. Is one of you more likely to start it? If the same person is constantly starting a fight there's something else going on. They are unhappy for some reason which may have nothing to do with you, stress at work, with family, money etc gives us all short tempers.