My heat pump is shot and is operating off the emergency heat. Can I just continue to use it this way or do I have to replace the heat pump?

Is your mystery system designed to operate as such

Stays on because it can not satisfy the tstat or continues to heat past set point

That is an enigma and it just makes no sense!!!!!!!!Could the switch involved be the culprit? I would check that out just to make sure. Try also checking the breaker by replacing it. Sometimes those things are guilty of crazy operation.

If your thermostat has an emergency heat setting this means you have a heat pump system.
Yes you need to set the thermostat to heat for heating the house.
The emergency heat setting should never be used unless your outside unit is not functioning.
The emergency heat setting overides the heat pump (the outside unit will not be energized).
The emergency heat setting will run the auxillary heat (heat strips) only.
When the thermostat is set in the heat position the heat pump will run to heat your house.
If the heat pump can't keep up (extreme cold) then then the auxillary heat will kick in to augment the heat pump until it satisfies the thermostat settings. If your room temperature gets 2 degrees below your thermostat setpoint, your auxillary heat will kick on. Example, say you are going to be gone a couple of days and you turn your heat down to 60 degrees. When you return and it is 60 degrees in your house, you turn the thermostat up to 70 degrees. Since you are turning the temperature up 2 degrees or more (10 degrees in this case) from the 60 degree room temperature then the heat pump will come on and the auxillary heat strips. The unit will continue heating like this until the temperature in the house gets to 69 degrees and the heat strips will turn off and the heat pump will continue to run until the 70 degree setpoint is reached.
Hope this helps.
SeagullAC

Heat pumps require a reversing valve for heat operation on the outdoor section, a control signal to a "O" terminal or orange wire is required for heating and cooling depending on the thermostat and out door unit requirements. The thermostat also has to be heat pump type for the system to operate heat for electric system with heat strip, two stage electric heating.

Have you checked the heat cycles per hour? In the tstat configuration?

The emergency heat setting on the heat pump thermostat is manually selected by the equipment owner. This is usually in response to a malfunction in the outdoor unit. Doing so locks out the outdoor unit. The indoor auxiliary heating system must provide the heat required. Setting the thermostat to the heat position allows the outdoor unit to operate. Due to the expense of electric resistance heating compared to the efficiency of the heat pump, repairs should be made as soon as possible.
MORE HERE

If you need further help, reach me via phone at https://www.6ya.com/expert/dan_73bbd84fe1d95b61

Switching to emergency heat will not defrost the unit.

If the Heat pump is malfunctioning, continue to use the emergency heat until the unit can be repaired properly.


At ten degrees, no heat pump can keep up., but it should be automatically bringing on the heat strips as back up heat. (same thing as emergency heat, it's just automatic.)

Either the extreme cold is overloading the defrost system, due to lack of properly operating exterior temp sensors or the defrost is malfunctioning,

the problem sounds like 1 or 2 things either you have a bad sequencer for the heat strip or the emergency heat portion was not wired up.