I thought that might have been it. The DHCP profiles didn’t exist last time I looked, but the static address profiles were set to auto start.
I noticed last night that the ethernet adapters changed, and the static profiles didn’t update to match. The adapters were named something like enp6so, but used to be enp2so, for example.
The DHCP profiles matched the new device names, and the static profiles were stuck on the old names.
Changing the static profiles to match the updated device names and deleting the DHCP profiles seems to have worked for now, but I don’t know why they changed in the first place.