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.

Does the furnace shut down when the thermostat stops calling for heat (or very soon after)?

Furnace

If the furnace does not shut down when the thermostat stops calling for heat, it's a problem with the furnace. It likely won't stop immediately, but should stop fairly soon after.

Thermostat

If the thermostat does not stop calling for heat once the set point is reached, it's a problem with the thermostat. Usually there is some form of indicator on the thermostat, that lets you know when it's calling for heat.

Read the manual for the thermostat, and make sure you're programming it properly.