Typically 1 of 3 things will cause furance fan to run continuously
1) Faulty Thermostat
2) Shorted T-stat wire(s)
3) Faulty timed delay relay
The timed delay relay is what triggers blower motor to start/stop
It can be a separate component (small box with 4 wires. DC power from FUSE to relay then to motor AND wire from t-stat to relay then to ground)
OR it can be part of the circuit board
Yours being a 2005 model......probably on circuit board which would mean a new board
And if it came on when you plugged into AC power it was getting DC power vai converter------so battery is dead (Or hopefully disconnected)
Is it time for your medication or mine?
2007 DODGE 3500 QC SRW 5.9L CTD In-Bed 'quiet gen'
2007 HitchHiker II 32.5 UKTG 2000W Xantex Inverter
US NAVY------USS Decatur DDG31