Thanks for your suggestion. I had already checked for old versions on the system, plugin paths, etc., but I did it again just in case I missed something.
There is only one flashplugin file on my whole system and it's in /usr/lib/mozilla/plugins. Opera finds it through its plugins path and I have a symlink to it for SeaMonkey (probably unnecessary). Previously I had tried copying the flash plugin to Opera's plugins directory and also putting a symlink in Opera's plugins directory. None of those helped. I also renamed my ~/.opera directory so Opera would recreate it. That did no good either, so I restored my ~/.opera directory. Opera does show Flash as installed.
I did a Google search and also checked Opera's user forums and there were no reports of this exact problem. So for now I think I'm out of luck. When I need Flash I'll use Firefox or SeaMonkey.