MySQL connections are automatically closed as soon as the script is finished, so you don't need to close them explicitly. However Q2A does close it anyway, just to make sure it's done at the earliest opportunity. See the bottom of qa-page.php.
Check your plugins are not making too many SQL queries. The more you are making, the longer the connection stays open and is not available for others.
Make sure you have QA_PERSISTENT_CONN_DB set to false in your qa-config.php. See my answer here for a few more tips. As I mention there we have some efficiency improvements coming in v1.8 so maybe that will help.
Otherwise... if you have a lot of traffic then there's not really much that can be done on shared hosting. Perhaps try shopping around for cheaper VPS hosting. There are also hosts like Linode and Digital Ocean where you can get great servers for only $5-10/month, but they are unmanaged and you need to install everything yourself.