I think that I have found the problem that I was having where it appeared that apache was leaking.
I had some mod_rewrite rules that forward request from where this blog was hosted to the new domain name.
RewriteRule ^/blog$ http://www.paulstimesink.com/ [L,R]
RewriteRule ^/blog/(.*) http://www.paulstimesink.com/$1 [L,R]
I changed this to:
Alias /blog /<path_to_LifeType>
When people browse the old url, the url that they will see in their browser will be the old one, but I think that is reasonable.
The reason that I think that this is the problem is that when using redirects for two hosts on the same machine, this causes double of the connections open at the same time. The first will be the original connection. Then the redirect, will cause the browser to open a second connection. Most browser should close the previous connection before opening the next, but some browsers may not do this.