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While the fan will run continuously with the fan side of the Fan/Limit jumped out the Fan/Limit is for heat only. Replacing it won't solve your AC problem. There is a fan center relay that controls the fan for AC.
Yes, it sounds like this in an old unit so I don't know the part number offhand. But it's a pretty common relay. If you remove the old one and take it to an HVAC or many plumbing supply houses they should be able to give you a suitable replacement.