JW, wow I’m sorry you have all this to deal with.
Trailmix said it well.
as you said, you can’t trust her. You’re right, you can’t.
To be clear, you are dealing with two separate issues regarding trust. Alcoholic lying is a hallmark of addiction. When we aren’t recovering, we’re lying. Usually about drinking, but the habit can certainly spill over to other stuff. If the issue is “simply” alcoholism, when we recover from that we stop lying. Notice I said “recover,” not “stop drinking.” Recovery as you know takes time and is more than just abstaining.
the cheating is a whole other level of dishonesty. cheating is cheating. If her alcoholism disappeared tomorrow, the cheating and dishonesty of that would remain if not also addressed.
I say this because people sometimes think that an abusive or unfaithful spouse does those bad things because of addiction. They may be made worse by addiction, but are not CAUSED by it.
good luck to you. I hope you keep building your support, there are a lot of kind people here. Take care.