Called back to the voting office today to get things finally straightened out, but there was some confusion at first. The lady at the office took a look at the file of the other person with my name, but the driver's liscence number was the same as mine, and the signature was identical.
She went off to look over the situation with her supervisor and called back a little later. It turns out this was literally due to clerical error: someone down in Broward County, where this other person had moved to, accidentally took my information and filed it in his folder.
Once we figured out the exact cause of the problem, we got the ball rolling to fix it. I should have a new registration card in the mail within the next couple of weeks.