I have this same problem with this poll thing now at the top of my screen on my home computer. Can't make it drop down even though I have voted. It's at the bottom of the screen on my work computer.
It is annoying. Did you ever get a fix for this?
No, I think it requires a FreeRepublic software fix. I'm sure John will find a solution during the next update.
The following is just my completely uneducated guess: I believe that your vote is not only registered on the FreeRepublic server but also sets some special cookie info on the computer you used to vote. (cookies in the internet world are just small packets of data that websites keep on your computer for future reference)
I also believe that the cookie info determines if the poll/vote box is displayed at the top or bottom of your sidebars. But the info on the server defines whether or not you get to see the vote prompt or the poll result data in the box. So when the server doesn't agree with your cookies strange things like this happen.
I'm guessing that the complication arose out of safe guards against people voting multiple times coupled with the new tracking of registered vs. non-registered votes.