We would not restore your database files. As mentioned above, you would need to let us know which tables you believe to be increasing in size incorrectly. Alternatively contact your hosting company to ask about this increase in size on your database.
Looking at your logs, you have quite a lot of logs relating to downloads, which look to be being caused by a 3rd party application. Try disabling that to see if your logs size goes down.
You do need to note here that if you start getting repeated logs, that table will indeed grow. 350mb is actually a very very small amount these days.