After blowing heat the fan won't stop, continue blowing cold air for prolonged period of time (30+ min)
The blower remaining on after a heat cycle can be a symptom of a failing safety sensor (but we don't have enough information yet to confirm whether that's happening here). The logic is that if the controller can't confirm the gas burner has shut down then the blower is kept on as a safety to avoid the furnace literally burning itself up. In this case the inducer fan (the large black rectangle in the middle) would also remain on.

You'll need to carefully observe the furnace and report back in order for us to help further.


Adjust the thermostat to call for heat
Turn off the power to the furnace for about 30 seconds, then turn its power back on. Hopefully there's a power switch (looks like a light switch) on the exterior of the furnace probably on its right side; if there isn't you'll have to have an assistant flip the circuit breaker.
Standing at the furnace while the power is switched on observe what happens. Make notes or record a video. Listen for clicks, notice when the main blower and the inducer blower come on (the inducer blower is the black rectangle in the center of the photo), watch for an orange glow to appear in the burner area and for the gas to turn on.


Edit the question to include the sequence you observe. We don't need to have the video uploaded (I'm not sure whether SE even supports that); a written description will be fine.