It looks like each of your mysql processes are using 80-100% of the mysql's resources, thus when you have many processing going on at one, it is bogging the down the mysql server and the server itself for others on the server.
Take a look at the processlist and see which queries are slowing down the server and log slow queries.
I just scrolled a little through the pages on git and it looks like the faucet isn't the best design. No prepared statements, no real db design. For example, index.php as the query "SELECT dispensed FROM dispenses WHERE email='$address' OR ip='$ip' ORDER BY id DESC LIMIT 1", but dispenses only has a key on id and no index on email and ip. With lots of entries, that will bog down the server. Might also be worth to try a switch from innodb to myisam.
Personally, I'd look at other faucets to be honest.