AFAIK, the only non-Radio Shack place in town is Electronic Surplus in
Mentor (used to be at E. 50th & Broadway until they moved a few years
ago)
http://www.electronicsurplus.com/
Good if you need a part fast and don't want to bother with a Mouser
order for a couple of diodes or whatever.
I've found it helpful to find what you need on the website first, and
bring a list with the part numbers, since a lot of their inventory is
not in the main area.
Sam