watchdog truncate or delete Watchdog table has become so big that, truncate cascade or delete command in the table is not working at all. 
The moment the server encounter a DELETE command in watchdog, it gets busy in executing that, all resources are being occupied by DELETE command and server gets very slow. The total functionality can be considered to be null.

Even vacuum full/ananlyze is yielding any result even after waiting for 1 or 2days.
You can use truncate command which is faster than Delete. And try using this command directly from Terminal rather than doing "clear log" option available in Recent Log Messages in Drupal 7.

You can find more details on Truncate vs Delete on here.

Hope this helps.