You can replace that Coleman Mach t-stat with a Hunter t-stat.
Hunter t-stat that uses an internal battery for the t-stat.
HERE are the direction/article.
Furnace fan continuously running is typically a problem with t-stat, shorted wires from t-stat to furnace or a faulty timed delay relay on furnace.
Furnace (and t-stat for furnace & A/C) needs 12V DC to operate.
Your converter is supplying the 12V DC power.
When t-stat contacts close....furnace fan runs 15-20 seconds to purge burner chamber then spark occurs. Main flame lights. When t-stat contacts open (temp set point reached) main flame goes out and fan runs until timed delay relay opens. It is a separate component that has DC power from DC dist panel (fused) and then goes directly to fan motor.
It is a normally open Relay. When activated closes, sending power to the Blower Motor. When deactivated allows Blower to run for a period of time to cool the Combustion Chamber.