I'm not sure you have a problem. Sounds like the fans and related relays are fine since jumping the two switches makes the fans come on appropriately.

Fans should come on low speed only when the pressure in the AC system exceeds a certain value. On my car, this only happens in hot weather, in stop and go traffic. In other words, unlike in some other cars, the fans do NOT automatically come on whenever the compressor is engaged.

You say the fans do not come on at high speed either. They should only come on when the coolant temp exceeds a certain value. In my car, they seem to kick in when the coolant temp shows about 100 degrees or a little more.

Unless I'm missing something, the fan behavior you've described sounds like it could be normal.

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Steve
'93 400E