You are certainly not alone. My family has a history of Interstitial Cystitis. Years ago I had trauma to my abdomen in a riding accident, and after I had some nerve issues and my bladder was SUPER sensitive. Since then, my bladder seems to have sensitivity issues related to my anxiety level, also related to the trauma.
I was diagnosed by a Urologist with Overactive Bladder (Urge related). Good luck finding a Urologist that is female. They are apparently all male. You walk into the office and there are like 30 guys all sitting around in the waiting room. I suppose none of them can pee. I see a female NP that works with a male MD. See prescribed a medication that relaxes the walls of the bladder, which helps. For me, the urges are completely related to my level of anxiety.
You sound like you have stress incontinence according to this link: http://www.webmd.com/urinary-incontinence-oab/womens-guide/urinary-incontinence-in-women-symptoms
That might give you some ideas of other things you can do that might help. It was helpful to me even just to read about how common it is. In the very least, the Dr. can do an ultrasound to make sure you are emptying completely, and check you out for abnormalities and such.
It is a totally embarassing and horrifying situation though!! I sympathize!