Have you access to your database? It seems as though if you can upload a new file and it works, but only once and if you change your password it works only once there might be a problem with the database.
If you can check the database and look into wp_users at the user_pass and make a note of the encoded password. Try changing the password again, check the password, then login and fail, check the password again. If we can clear the user name and password from the problem it might narrow down the search.
Thinking about it if you can use your ftp to check the contents of wp-admin after you resert and see if any files change data/time. I am wondering if you might have a compromised system, but this should show any changes not made by yourself.