The system wasn’t exhibiting the signs that @sam described.
du postgres_data shows 9GB, so maybe the database is just too big? But it was working fine on a 4GB droplet before this problem started.
Load average is staying below 2.
Maybe it’s bad luck and that server at Digital Ocean is broken in some way.
@digitaldan, you might file a ticket with them and see if they know anything like the SSD is failing or something. Or some other user on that machine is breaking it.