I used to have really similar fears! It took me about 2-3 months to get over it, and I am over it now because DD has never had an accident in public.
I would say that if you make an effort to get your daughter to a toilet as soon as she says she needs to go, then you should be fine. This will still be longer than it takes you at home, e.g. maybe you will have to walk to the other end of the mall to find a toilet, but chances are she'll be okay.
I agree with PP - take the potty with you! I certainly have done for the park, the beach, and long car trips so we can just stop on the side of the road. I don't really do this anywhere where there is a public toilet available like the mall or somewhere.
I guess you just need to bite the bullet and go out - it's not likely that you'll get over these fears if you don't give her a chance! Just make sure you're well equipped with a full change of clothes for both you and her. (I have been weed all over and didn't have a change of clothes with me, luckily it was at a public swimming pool so I don't think anyone really noticed that my clothes were wet!!! Although it was still really gross.)