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What did the B and C wire go to on old thermostat and what brand heat pump do you have outside?
C(Common) is not used on the RTH6350.Screen is powered by the AA batteries.not 24v from the unit.
If you had a B wire and a C wire, the C will be taped off and B will go to the O/B terminal.
But as far as display nor being lit up that will be a battery issue.