The same thing happened to me for about the first three weeks of sobriety. I'd go to bed early but when the alarm went off I felt like I hadn't slept at all. I'd also fee like I was in a dense fog for the first few hours after I woke up. I don't know why it happened but the good part is it gradually got better and now I'm actually a morning person! At a minimum I'd go to the doc and have them do a thorough blood test just to make sure there isn't something that might be causing the morning fog. Then again I err on the side of caution these days.
Hang in there!