Drunkenness is definitely not moral turpitude, so you didn't need to apply for a visa but could truthfully have ticked the no box on the VWP form. (The London Embassy advise you to apply for a visa so that THEY can make this decision, and charge you for it.)
So there shouldn't be a problem with a visa, except....
the US are clamping down on visa applications from people with a drink problem. You will have to convince them that you don't have one. Denying that you were drunk won't work - they will take that as evidence that you DO have a problem and won't admit it. A one off lapse is the best line.
So I wouldn't like to predict how things will turn out. Sorry.