A technician will usually check refrigerant pressures, the electrical parts at the outdoor unit, the indoor coil and blower, and the condensate drain. If refrigerant is low, expect a conversation about finding the leak rather than only topping it off, since refrigerant work belongs to trained technicians.
If the system needs replacing, the Building Department at Town Hall on Central Street handles the wiring permit, and wiring inspections are booked by phone in a short morning window Monday through Thursday. Near the harbor or Crow Point, ask the contractor to set the new outdoor unit's height with your flood zone in mind; the town's flood map page explains which FEMA maps apply.
Many owners replacing central air in Hingham also price a heat pump that heats and cools. For oil, propane, or electric-heated homes, that can mean HMLP's heat pump program, while homes heated mainly by gas would look at Mass Save.