Welcome to the Healing Heart Community, where we all know the pain you feel. I am very sorry you are having to deal with the betrayal of the person you love.
Please understand that her inability to be faithful is about her and not you. No matter what you do, you cannot make her be faithful. People cheat because of a flaw inside them and not because of what happens around them or to them. You cannot make her faithful, you cannot make her unfaithful.
You ask if you can trust her. Go back and read what you wrote several times. If this were someone elses words, would you tell them they should trust their GF?
Can people change? Yes, they can, but they have to want to and recognize the problem is within themselves. Your GF always has some excuse why she engages in inappropriate behavior to down right infidelity, repeatedly. If drinking is her problem then she shouldnt drink. I know that my husband would purposefully get drunk before meeting with OW so that he could blame the liquor. Being drunk is not a reasonable or valid excuse, and should not be excepted as one.
When we love someone, we all make excuses for behavior we know we shouldnt. We want so much to believe that things are not as bad as they seem. We love/trust blindly, because we think this is one of the ways to be faithful to our partner. Loving/trusting blindly is a form of denial. After a loved one has been unfaithful we have to learn that we have to find a new way of loving and trusting, with out blinders on. You have to see your GF for who she is. It is not pleasant, I know I had to see my Husband for who he was. If your GF is sincerely remorseful she will be completely transparent, she will willingly go for a lie detector test if you ask, (I do know people who have done this), she will be accountable indefinitely, and not claim that she has a right to privacy. She will seek help through a therapist and learn why she has treated herself and you so badly. The list goes on. Then after a couple of years of constant changed behavior you will find you begin to trust again.
Now this is just my opinion but I think it would be a good idea to put off marriage plans for a couple of years until she has constantly proven her fidelity and gotten professional help. Things get more difficult after marriage and children.