ARE YOU USING A ROM? If so, that's your problem right there.

If you're playing the actual SNES cartridge, or if you're playing the PS1 version, I have one other suggestion. If you want a secondary means of switching your charaters, walk around on the world map and push the "Y" button (SNES version) or the "square" button (PS1 version). It will bring up a character switch screen.