Your site just may be query heavy. I have seen this a few times. You probably have logging on in WordFence and your site is being hit by bots causing WF to drain your available connections.
You need to stop the request at server level before they reach PHP so you should look into something like ModSecurity for apache. PHP scripts like WP are awesome at what they do but they require heavy resources when put to the test. I would also look into query caching for the MySQL side of things.