You cannot sign out! It's not a bug, it's a feature
Actually you can change your password via your pc and still be logged in with your smartphone. Dropbox doesn't care(!) Normally Dropbox should throw you out, when password has changed ...
I can't find the "settings-sign out" option anywhere. Instead, all I'm able to get to is settings-- accounts-- dropbox, and the only option is "remove account." If I go through dropbox-settings, then the only option is "unlink device from dropbox." I can't find any way to just plain sign out. If anyone knows exactly where the "sign out" option is located, PLEASE share, because I don't know how anybody is finding it.