OP, firstly, you are not alone, around 1 in 4 people have some fear of flying.
Secondly, and importantly, the fear and panic you feel is entirely understandable - you are in a situation you have no control over, you are flying - something humans have not evolved to do (unlike birds)- so your body and brain are telling your mind there's something wrong!
Panic is just a natural reaction we have when in a fearful situation, a fight or flight (no pun intended) reaction to save us from danger. The problem is, you cant really do either on a packed plane, so the attack continues.
There are many books on the subject, one recommended by BA on the Flying with Confidence course (which I've been on) is Living with Fear by Isaac Marks. BTW, I have no connection to him, or BA!
You may find understanding why you are having a panic attack, and what one is physically and emotionally may help you manage. There are breathing exercises you can do to help, and as others have mentioned your GP may prescribe beta-blockers, Valium, anti-depressants etc. but these many not be helpful if you have to drive at the other end, look after others etc.
PS. as I said I went on the BA course, found the lectures, talks etc very helpful, but couldn't actually get on the flight at the end. It's something I still need to address. 