The unit is blowing cool air when it is turned on heat

https://learnmetrics.com/air-conditioner-not-blowing-cold-air/

This may help you fix your problem.

If your system starts to blow cool air when your Nest thermostat is in Heat mode (or if it blows warm air when its in Cool mode), change the settings for orientation of your heat pump. The instructions depend on which thermostat you have

The cold that you describe may be just unconditioned air? If you are certain you can hear the comp operating for a couple of minutes, and then the heat starts, I would guess it is a reversing valve issue if it's a heat pump and it probably is. Or the control board acting up. I've had a lot of issues with the boards in the Amanda PTAC units. I can't remember how to configure it, but if you Google "Amanda ptac configuration", there is a lot of info online. May try a reset method first. Hope this helps!

Sounds like wiring issue to me. I've delt with units like this and there is a wire lead for fan only and a separate one to run the fan when the unit is cooling or heating. I'm only speculating here but I would bet money that this was improperly installed.

Dont get discouraged though because it sounds like its working (for the most part) if you feel up to the task find the manual and check your connections. Otherwise hire a professional.

model number? Heat pumps tend to blow cooler air than a space heater unless in money-gobbling mode.

Short of gas. Time to call a technician

it sounds like unit could be kicking out the low pressure switch or if it is a electric furnace it could be the sequincers

Turn thermostat switch to 'cool' setting. It's a 'heat pump' and is in the heat mode.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/craig_3fa289bf857b1a3c

It sounds like you have a unit called a heat pump. When a heat pump unit is in cooling mode the outside unit blows hot air and the inside unit blows cold air. If the outside unit is blowing cold then the indoor unit is most likely blowing room temperature or warm air. It sounds like the thermostat could be controlling incorrectly or you could have a failed reversing valve which is the device that switches the unit from heating to cooling mode.