Cold air coming in
Can I close off the passage to the cowl and leave the door open on the plenum to recirculate the heat or am I missing something
That is clever! And it would work!
I have seen some clever cowl covers. Clear bra is probably the easiest.
Your car MIGHT have a valve that stops the flow of hot water???
Or the heat door is not opening correctly or enough.
You say you have heat for about 5 minutes but it gets too cold in the car while driving but have heat from the defroster. What kind of airflow do you have?
Have you measured the temp of the air?
Do you know if your diverter door is working correctly?
When the controls are in "Def" little to no air will be coming from the floor vents.
Another thing could be the engine thermostat. If it is too cold the car might never heat up properly. Just had that issue on my Jeep GC. and an old Chevy I had.
There could be debris blocking the air flow through the heater core, or debris in the heater core.
My convertible 67 dart. With the top up and in 20 deg weather here in Northern co has no problem heating up the inside of the car. I also remember as a kid riding in the car never feeling cold.
I think you have some detective work in order.