It's not normal to start dreaming as you fall asleep ? I thought that's how you know you're falling asleep? That you're starting to see the dreams?
On the other end though I sometimes wake up in a state where I sort of seem to know I'm awake but not really awake yet. I guess that's still dreaming. Sometimes I try to control the dream but it often fails because I actually move my real body and I wake up and I have that "daaang, I wanted to keep dreaming" sensation. But then it's too late.
However, as hard as I try, I can't remember more than the fact that I was in that state even just hours later, never mind now. Except for very few times. Like I still have one specific split second memory of a nightmare I had when I was a kid (like 30 years ago) and woke up from it. But I can't remember anything but some sort of vague "brown paper lunch box head monster" thing.