Thanks very much for this answer. It will help me have a more constructive dialog with my hosting service. They offered to migrate my sites (I have a second similar site for a different alumni body, which is having the same problems) to a different server with less traffic. I will refer them to the article you referenced. I will ask them to run the commands:

echo 8192 > /proc/sys/fs/file-maxecho && 32768 > /proc/sys/fs/inode-ma
+x
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ulimit -a
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ulimit -n
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ulimit -H
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I would also like to try inserting

warn `ls -l /proc/$$/fd/` . ' ';
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into the 3 most popular modules in the script. Finally, thanks for the suggestions on not capitalizing Perl. I had no idea. The last thing I want to do is offend the very people from whom I need help.