hi and welcome
You can't give your wife anything. If she wants a divorce she can have one. In learning about alcoholism I found out that being unfaithful is something people do and it is not a feature of being an alcoholic so if you are expecting her to change if she stays sober you maybe disappointed that that side of her doesn't change.
I don't want divorce.
Maybe you need to work out why cos from what you have said their is precious little to save in your marriage, your wife is having affairs and has told you she doesn't love you. Her behaviour is effecting your children. Alcoholism is progressive and she will only get worse. Is this what you want for your life and your kids?
So yes I'd "give her a divorce". I divorced from my alcoholic husband 4 years ago and life has been much better. Learn all you can from the stickies at the top of the forum about alcoholism. You will see that much of her behaviour is her and nothing to do with her drinking.