I have this same problem. A good place to start is with an internal medicine doctor, get the usual labs check...a chest xray and EKG. The expensive thing you may want to look at is an overnight sleep study. If your income isn't too high, try to find a hospital that does charity (most universities or county hospitals do)
This could be a sleep disturbance; more specifically, physical obstruction to the movement of air into and out of your lungs, which is called obstructive sleep apnea. It could also be due to irritation of your vocal cords from gastroesophageal reflux disease (GERD) or, given your seemingly faint pulse, it could be caused by an abnormal heart rhythm.
Not having good health insurance is a terrible problem and some of the testing for what you are experiencing can be quite expensive. You say that this happens, "every once in a while". The frequency is important in determining whether you should try to find the money to be evaluated. Your age, the duration of the attacks where you experience the "feeling like I can't breathe" and other associated symptoms would also be important in making that decision. Something like this that happens once a year is very different than its happening 2 or more times a month.
The best advice we can give is that you arrange to see your primary care physician, sit-down with him/her, present the facts as best you know them and get a professional opinion.