A bad fan motor would have nothing to do with an overloaded thermostat- depending on the age of the unit, it could have either a fan relay or fan control circuit board that the thermostat controls. Smoke pouring out of the thermostat can indicate the anticipator is set incorrectly- if it is set way too high, it will smoke. If the thermostat wires are fine, I would try simply connecting them together.
Not sure about the light dimming, but given the thermostat wires are so small, there cannot have been to large a draw through them. It could be 2 problems- thermostat and blower.
-- Chris Bryant