It sounds like the fan coil may be being energized by the G circuit on your thermostat instead of the w or y circuit which should energize the fan for heating. Take the g wire loose at fan coil probably wire nutted and with it loose call for heat and tell me if the fan still comes on
what turns the emergancy heat? Besides the switch at the thermestat.
Second stage thermostat setting most likely turns on the emergency heat if you tighten the differential bands you might help that way
Fan would be energized by R closing to W1 for heat and R to Y and G for cooling
But R to G will energize fan on with out any heat or cooling