This is one of those problems that many of us face at one time or another: the psychological difficulty in acceptance that lies at the heart of "dealing" with MS. In my view it has two main components: coping with changing your own image of yourself, and imagining how others are going to see you and what they'll think of you. Because you say you'll feel embarrassed I guess you're very concerned about what other people will think, so I suppose it might be useful to ask yourself what you think when you see people using some kind of walking aid – chances are it's nothing more than " oh look, that guy has something wrong with his legs" if, indeed, you give it a moment's thought at all.
Also, it sounds as if the falls you have had are pretty serious and happen fairly frequently. When I was a bit more mobile I used to fall often and it was the fear of injury, and the dread of falling somewhere and being stuck without help, which made the use of a walking stick a much easier option to take. I felt much less embarrassed if I tripped and fell while using a walking stick, (and other people could plainly see I had a "reason" for falling and not jump to the conclusion I was drunk), than if I had just simply sprawled onto the floor in the middle of the supermarket.
I know the kind of cane you're talking about but I'm afraid I don't know what they're called, (it's probably something different here in the UK anyway), but the thought occurred to me that they are very often given to people with a broken leg, so if you find yourself wondering what other people are thinking, try telling yourself that they'll just assume that.
I hope that helps a bit,