What to do about furnace that’s drawing gas odors into the room instead of heat

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.

If the furnace is constantly blowing cold air, it's likely one of two problems. The first is easy to check and fix, the second will likely require a service technician. Start by checking the thermostat, to make sure the Fan setting is not in the ON position. If the fan is set in the ON position, the fan will blow air continuously. Make sure the fan is set to AUTO, which will allow the furnace to control the blower.

The other reason a furnace blower will run continuously, it that the high limit switch is dead or dying. If the high limit switch fails, it will "fail open". This causes the furnace control board to think the furnace has overheated, so in response the control board turns the fan on to clear the heat. The most common cause of high limit failure is constant overheating, which is usually brought about by dirty filters. You'll likely have to contact an HVAC Technician to replace the high limit switch, since it might be difficult for you to locate the part yourself. In the future, make sure to check your filter and change them as often as needed.