So to clarify, the blower will run only if the thermostat has the blower set to "on" and not when it's set to "auto"?
The problem is either the thermostat, the thermostat wiring, or the furnace controller.
What I would do is disconnect the thermostat, in case it's doing something strange, and jumper the Yellow to the Red wire (24V RC to Cooling Call). If the system operates normally in this mode, then it's the thermostat. If not, then the furnace controller board is not working properly.
If yours is a more complicated system, like a heat pump, then there will be more wires and the wire colors may be different.
Be sure you understand what wires do what, turn off the furnace breaker first, since 24V hurts a fair amount, and you could wipe out the furnace controller if you shorted it. If you are not comfortable working with this sort of thing, then a service call to a HVAC pro would get it fixed very quickly.
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