HVAC blower won't cut off

[zoomajor]

Hi, folks. I've got a 20 year old unit that needs replacing, but I need it to last 1 more season. The unit heats fine(A/C needs freon, but it's heating season), but the blower runs continuously as long as the switch inside the closet is on(turns off when this is off). The thermostat seems to function OK, as it turns on the burners when temp and setting match and cuts off flame when desired temp reached. Blower remains on whether cooling or heating. There is a solenoid looking device atop the toggle switch box in the closet. I'm wondering if this is what could be the problem.Any feedback is appreciated.

Thanks,
zoomajor