The server started behaving normally on Monday though I had done nothing. I don't have any definitive answers for this but here are my theories...
Theory 1: The host machine in the AWS infrastructure had problems or was overstressed over the weekend. I don't know enough about virtual systems but I am constantly told that your neighbors can impact performance one way or another. Bear in mind I never got any error messages of any kind.
Theory 2: Since it started on Friday at midnight and seemed to end on Sunday, perhaps AWS engineers were doing something awful. Kind of goes with theory 1.
In order to do "something", I'm going to upgrade my AWS machine and reinstall AwareIM. This does 3 things... (1) moves me to a new neighborhood in AWS, (2) puts more juice on my machine, and (3) reinstalling software is always a good shotgun approach to reset the baseline. My next step would be to move to a dedicated server, which Support and others have recommended. But if I do, I should probably move the database as well, which is somewhat harder. I prefer to stay in AWS. If you have more advice about the value of a dedicated server, or the woes of a virtual server, please let me have it.