Hi Sandy,
I don't know if it is "normal" but I do know I have felt that way. When I was feeling panick alot, I hated when my husband would say he was going to bed, cause number one, I was now alone if I stayed up, and number two, I was waking up now and then like you say, racing heart and short of breath. I would even ask him "Do you feel like there is no air in here" but he didn't. My counselor said, and from what I have read, anxiety can do that to you too. What I did was buy a couple of great relaxation cds and listed with headphones. I did this everyday for a couple of weeks and now rarely need or even want to. When I was really bad, I would even wake up about an hour and a half when the cd was over, but instead of panicking like "great now what do I do," I would just hit play again, and once again would drift off.
Give it a try, but most of all, don't worry too much if it is anxiety. I say this meaning, you have been to the doctor, or at least know you are definitely during the day struggling with anxiety.
Bonnie