You're not the only one. I used to (I have a stoma now). There are many different causes, including (as you mentioned) damage to nerves or muscles as a result of trauma like childbirth or diseases. Crohn's does not affect the nerves in the way that some illnesses do. It is more likely to cause incontinence due to severe, urgent diarrhoea that does not give you enough warning to get to the bathroom. As you are not always feeling an urge at all before it happens, it may be that the Crohn's has caused some damage to the rectal area, maybe a tear when stool can seep out, or an abscess or something similar that could leak (it may be mucus leaks too), or, when Crohn's causes frequent diarrhoea over a long time period, the stress of this on the rectal area could cause a common problem like a prolapse or haemorrhoids that can result in leaking stool or mucus.
I'm not sure how the affects of fibre would affect any of these conditions, but fibre obviously has effects on frequency and consistency of diarrhoea.
A digital rectal examination is a simple test a doctor can do to assess you for the colorectal abnormalities that I mentioned above. A sigmoidoscopy (like a colonoscopy but less major as only the end of the bowel is accessed) can give a more thorough examination. These conditions are generally treatable. If there are no abnormalities, then it's a matter of getting the flare under control and stopping diarrhoea.
You will need to see a doctor though. I know it's embarrassing but it's the only way to know what's going on and get treatment. You've not done anything wrong. It's a genuine problem you are right to be concerned about and no doctor should blame you for that.