After some research, I determined that a lot of the downtime that had occurred in the past week may be related to a lack of memory on the new dedicated server. I could have spent a day debugging this and trying to reduce memory usage, but as I told Rize in December, it would make more sense for me to just work a day of overtime instead and have enough money to pay for two years of memory upgrades.

So that's what I did. Assuming that memory was the problem, there is now a definitive resolution as well as having a more capable server in the end. Hopefully this change should result in the elimination of out-of-memory errors.