'89 fan relay? Fan on all the time

Have a new-to-me 89 Pao.
The fan turns on any time the ignition is on. Found that the fan is hot wired to a side of the coil, so obviously the thermostat switch or something else is wrong.
Any tips what to check?
Where to find the thermostat switch?
Should I just rewire in an aftermarket thermo switch?