Ok, I am an electrician. I do residential service on a daily basis, and have been for over 4 years now.
I am impressed with the amount of knowledge you guys have on the subject!
Yes, by code, your A/C AND Dryer need to be on dedicated circuits. The dryer pulls off of two phases, which is what makes it 240v, and why it's a two pole breaker.
If you are going to go ahead and replace the breaker, which is a good suggestion at this point, please please please turn off the main breaker. If you are living in an older house that does not have a main breaker, I would highly recommend calling a pro. If you want to go ahead and do it anyway,and assuming your panel is in the basement, be aware that any concrete that you touch is DIRECT GROUND, meaning if you touch and part of the panel that is energized, you have the potential of drawing the ENTIRE capacity of the supply transformer through your body. wear shoes.
I would also highly recomend replacing your GFI breaker, it has tripped many times, and may not be protecting you as you would hope. Being that you and your family are swimming in a potential difference of 120v, it's a definite safety issue.
do you have any room in your panel left? Most likely not, that is the best reason (though highly illegal) that someone would put them both on the same circuit. if you do, you simply need to get another double breaker that matches the max breaker size, listed on the side of the AC unit. it's ok to go a little smaller, but NOT bigger. Follow the AC power line and see where it goes. if it goes to the panel, you should be able to make somethinhg work. If it goes to a junction box, you will need to run another line out of that box to the panel.
good god, I'm rambling....
I'll leave it at that for now... any more questions, do NOT hesitate to ask I am here to help...
no charge. ;p