The radiator fan will turn on after pressure builds in the system, not as soon as you turn on the AC.

Try unplugging the pressostat on the low pressure line and use a paperclip to jump the wires. If the compressor does not engage, you could have a bad pressostat or a worn clutch.

To determine if the clutch gap is too great, I take a broom stick and carefully tap the clutch face and see if it engages. Lots of moving parts, so watch yourself. Smarter people use a feeler gauge and check the clutch gap. If the gap is over .25, then make 3 shims and glue them to the clutch face to decrease the gap.

The pressostat can be replaced as there is a schrader valve between it and the line.
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My back feels better when I sit in a Volvo seat