That makes it sound like the thermostat really was defective, and that replacing it has solved that part of the problem. It sounds now like the hot water from the engine is not circulating through the heater core.
I have been told that some engines (Pontiac) have a tendency to develop an 'air lock' in the cooling system, which prevents the hot water from circulating. I've never heard of this happening on a Contour though. I would try to figure out if the hot water really is circulating from the engine to the heater core. I have no experience in doing this, though, so I'm only thinking out loud. I'll check through my service manual and see if I can find anything. Good luck.
95 Contour Zetec, ATX (wrecked, sitting in the barn)
95 Mystique Zetec, MTX
2000 Taurus SEL 3.0L Duratec
1994 Crown Victoria LX 4.6 V8
1993 Chevy Silverado 4.3 V6
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