The picture is of all the hosted domains and what is being used on the
domain. Nothing too unusual, and none of the sites are heavy traffic
sites anyway.

I've taken all their suggestions and already moved one of my sites
that was identified as a "problem" but that didn't seem to have helped
either. Unfortunately, I can't talk to them on the phone and I'm stuck
trouble shooting through a ticket system. Can't the logs and other
information generated by the server better direct me to the problem?

My intention now is to migrate to DreamHost but that will, of course,
take time. One of my sites is a class I'm teaching and this disruption
is not going to help student success this semester.