Water – No water pressure in heating circuit

Can you look at the furnace during the time it is blowing cold air and see if the flame is on?

The thermostat has the ability to control the fan, but typically the thermostat sends a "turn heat on" signal to the furnace and the furnace controls are setup to turn the flame on for a X amount of time (to warm up the air) and then turn on the fan once the timer expired (so the air coming out of the vents is not cold). When the heat is turned on is the flame and fan going on at the same time?

Not sure about your furnace, but some have a couple of LEDs that are lit up and can be used to read a status or error codes. If you have the manual you could try to decipher the code while the cold air is running and see if it gives you a clue.

I would start by trying to figure out if the problem is with the thermostat or the furnace by installing a new thermostat and seeing if the problem goes away. If the problem is still there you can just return it. I had a furnace issue where it was going off randomly and would work in heat mode but not cool mode. It turned out that the thermostat was bad and sending intermittent signals.

Turned out to be the High Limit switch. The switch had burnt out, which caused the burner not to fire. Turns out when the limit switch senses the heat box is too hot, it shuts down the gas and forces the blower to continue so it will clear the excess heat. When the switch failed it always told the system that the furnace was over heated, so the burner would not come on and the blower would run constantly to try and clear the excess heat.

To trouble shoot this issue I used an ohm meter to test each sensor, to make sure it was open/closed as it should be. Another way to test would have been to use jumpers to bypass the sensor all together, until the faulty sensor was located.