I apologize if you came to this site and it was slow.
I just upgraded the VM that runs this site to address the problem. I am hopeful there will be fewer problems now.
It appears that the old VM was memory-bound or near-memory-bound a lot of the time. This was likely leading to the CPU problems because of paging.
Now the VM has room to breathe on the memory side and I believe that will resolve the problem. I have no proof that memory was the actual problem, only a correlation between slow times, higher memory usage, and CPU spikes. Time will tell if this resolves the issue.
In the meantime, paying a few more bucks to see if things get better seemed like the right thing to do.