All I can say is that Geeklog sites like groklaw.net (which is slashdotted every other day) seem to be handling the load just fine, so I have to assume it's the way your hosting service has set up MySQL that is the problem here ...
The way Geeklog uses connections to MySQL should actually be pretty efficient. To quote from the
mysql_connect manual page:
If a second call is made to mysql_connect() with the same arguments, no new link will be established, but instead, the link identifier of the already opened link will be returned. [...] The link to the server will be closed as soon as the execution of the script endsbye, Dirk