I was just like Zebra.
I noticed I'm much more sensitive to stimulants now. I went from drinking 2 pots of coffee to none in the last 9 months. Now, if I even have a cup in the evening (such as at a meeting on a night that I'm very tired) I will be up until 3 am. Last Friday I was up until 6am.
I'm sober 9 months (-1 week) now and it is still a challenge at times. I used prescriptions (from my doctor) for a while - and they worked byt I was developing a tolerance - and we all know what happens next!
So I've quit sleeping meds in lieu of natural sleep aids, herbal teas, and really good sleep hygeine - that is, keeping a schedule. This is not unlike what most normal people do.
I also notice my ability to sleep is directly proportional to the amount of physical activity during the day. I work in an office and have a long commute so that is one of my challenges. Even going to the grocery store or Target and the like can help. Park away from the store and walk the entire store -even if it's just to get milk.
In worst case, you'll likely find that you still feel better with very bad sleep than you did with little sleep while drinking.
Best of luck to you.