Overheats when idling

Hello, this definitely does sound like a cooling fan issue. You'll have to use some diagnostic equipment to find out if the fan is being commanded properly and not functioning, or if the command is not there. Check for power at the motor when it should be running at the very least. https://repairpal.com/repair-shops/mercury-repair-and-service-in-raymore-missouri

Possible open high amperage fuse : https://www.autogenius.info/mercury-cougar-8th-generation-1999-2002-fuse-box-diagram/