Hello FriendlyGuy and Welcome!
I understand the feeling when you find out someone you love and thought loved you was unfaithful. My ex husband had an affair.
One thing I truly believe....alcohol is not an excuse. There are plenty of alcoholics in active addiction who would never, ever cheat. There are some who say they would never cheat except have while using alcohol.
Whether or not you decide to pursuit the relationship or leave is really up to you. If I ask you what you want (other than for none of this to have happened), what do you believe your answer to be? If you have ever given any consideration to the type of long-term relationship you want, what does that look like? Does this relationship meet any of those needs?