Since publishing this post, we have moved the site to Dreamhost/DreamPress. I've been very happy with the results. All of my previous issues have disappeared and the website as a whole is much more responsive. It's interesting to note that just after we left HostGator, my account was one of the accounts on the list to be migrated over to a new server box. Also, a Level 3 tech has been in touch with me today to try and figure out my WordPress issues but it's all a moot point now.