If you have changed your password that many times without anyone knowing the new passwords, chances are it's going to be on your computer. Keep in mind that Avast Free is not going to be very effective in finding the threat, as it is not only a free edition, but Avast do not make the most reliable of security software.
To try to fix the problem, I would perform the following:
- Download and install Spybot - Search and Destroy (this is 100% free, with no spyware, no adware, and very effective)
- Run the program by right clicking the shortcut on your desktop and clicking "Run as Administrator" (if you have Windows Vista or Windows 7)
- Click the "Immunize" tab, and immunize your computer
- Once complete, click the "Search & Destroy" tab, and click "Check for Problems"
This should hopefully find your problem and allow you to eliminate it from your computer
- Once the scan finishes, if there are any threats listed (in red), ensure they are selected, and click "Fix Selected Problems".
- You may be asked for a restart. If so, then confirm this, and make sure you've saved and closed anything precious/sensitive
I hope this helps you. If it doesn't, it could either be some serious spyware on your computer, or it could be on another device which has access to your email account somehow. If the problem persists, it may be easier to get another email account and see if that solves it!
Good luck!