When the thermostat is closed, water still circulated through the heads and intake via the bypass circuit (through the heater core too). When the engine warns and the thermostat opens, water flows through the radiator and the thermostat closes off the bypass flow. The coolant either flows through the radiator or the bypass, depending on thermostat position.
The thermostats for these are somewhat unique, Motorcraft RT-1039, or Ford E3TZ-8575-C. I'd go on eBay and compare the one you have to that. If you have one that looks like RT-252/D4ZZ-8575-A, it won't close off the bypass correctly.
I have a NOS RT-1036 I can send you, if needed.