If you believe your drinking played a part in this isn't it dishonest to give them only partial information to base their decisions on? At this point don't you owe them (at last) to meet their minimum expectations of responsibility and honesty from their employees?
Without doing that and letting them know you've taken actions to remedy this problem you are still operating on a sub-par level as an employee. By coming clean with them now and staying that way you approach the standards they hold for those they've hired and trusted to perform for their business.
You'll need to continually meet the conventional standards at some point, in this job or another. Putting that off serves no purpose I can see.