Have you checked any logs or analytics related to your site traffic? If everything in the code is fine then it seems an dramatic increase in connections would be related to an increase in page loads (or at least the scripts executing MySQL).
I'm also not sure if the standard PHP error_log file tracks the 'too many connections' error as well or not, but if it does it'd be worth a look since it should tell you which script the error occured on. It's just a suggestion since you can't get access to the MySQL error logs. If you could find out which script is creating the issue then you'd at least have better insight as to what could be the cause (for instance, a script that might be getting exploited by bots/etc. that gets run repeatedly opening a large number of connections at a time).
The only other thing I can think of (now that I re-read your original post) would be the method that you use to connect to MySQL. If it's not mysqli or PDO then perhaps this could be contributing to the problem. It's possible that perhaps HostPapa updated their version of PHP. The current version (5.5) has moved past the original MySQL methods and so you must now use mysqli or PDO.