There could be an issue that pages are taking a long time to generate, so that the database connection for each page view is open a long time, leading to high concurrent connections.
You can find out more by changing this line in your qa-config.php file:
define('QA_DEBUG_PERFORMANCE', false);
... to ...
define('QA_DEBUG_PERFORMANCE', $_SERVER['HTTP_HOST']=='1.2.3.4');
Replace 1.2.3.4 with your IP address, which you can find out from www.whatismyip.com. Now as you browse your site, you will see information at the bottom of the page showing how long it took to generate each page, and how that is broken up into different stages and types of operation. Have a look and see if one thing really dominates the processing time.