Does having a thermistor on all the time use a lot of current?
Depends on the resistance of the thermistor and the voltage passing thru it.
You will have the thermistor in a voltage divider: Voltage divider - Wikipedia
So add the resistances in series and calculate power with P = V*V/R.
You should be able to share the thermistor but I'd use a cap to ground from the thermistor/divider output to reduce input impedance.