Try re-installing it, but first see if it's running by uploading something either from your phone, or use the website. If you don't see the notification that should be in the top right of your screen in a couple of seconds, go ahead and reinstall. It shouldn't touch any of your Dropbox folders.
One more thing: On update to 13.10, at least for me, it disabled the ppa's for some of my packages. Go into Settings > Software & Updates > Other Software and see if something's disabled that shouldn't be.
Try updating manually. Open up the terminal and type "sudo apt-get update && sudo apt-get upgrade" to update.
I had 13.04 from scratch, then upgraded to 13.10 and added Dropbox the same day that 13.10 came out. I didn't have any problems, so just re-install maybe?
If the above doesn't work, and if you haven't already, do what any phone person would tell you; restart your computer. Try opening the terminal and doing a reset that way by typing "sudo shutdown -r now".