1993 Pontiac Sunbird Always running cold

Heater problem
1993 Pontiac Sunbird 4 cyl Front Wheel Drive Manual 134k miles

I've had the car two years now and still have not been able to figure out the problem. Makes me sound like a moron, but I really haven't been putting too much time or money into it because it is just my beater car. The gauge never moved, after diagnosing that, it was the sending unit (gauge checked out). I replaced that, but the car still ran cold. According to the gauge it would never clear 120 (and thats in the summer) and could watch the gauge fluctuate as if the thermostat was opening every 30-60 seconds. After replacing the thermostat, still no change, so I put another one in, still no change. I took a Snap On temp gun and checked the temps around the engine. The sending unit was the temp the gauge was reading, and the thermostat housing never got more than 120, so what the gauge was telling me is exactly what was going on.

Most recently I took the cap off (engine cold), sucked some coolant down and started it up. Coolant is pushing past the thermostat. Not around the outside where the O-ring is, through the valve of the thermostat. WTF. My only thought to the cause of this would be if the heater core were plugged, not allowing flow for the water pump while the thermostat was closed. I flushed that out, ran the garden hose through the heater core for a good 10 mins on full bore. Filled her all back up and still not change. I installed a plate a couple inches in front of the radiator (cardboard just doesn't hold up to the elements). Sitting around idling (I know, that plate ain't making a difference if you ain't rollin) it never gets above 120. At least now in town it gets somewhat warm, which gives me a little heat.

I was a mechanic for 5 years, after getting laid off from 3 different dealerships I took up a career in metal fabrication. I never once came across a coolings system running cold that a thermostat didn't fix. It doesn't make sense to me, if there is a problem with the cooling system, it runs hot, not cold. But seeing the coolant push past the thermostat tells me somethings up with the flow, but what would cause it to push past the thermostat?

Thanks,
Zach571