Can someone with an E-Bodies and No AC answer a question for me before I connect my heater core please?

When I connect this heater core back up, am I going to warm up the inside of the car every time I drive?

Before, I rebuilt the heater box, the interior of the car was like an oven every time I drove it. Now that I have rebuilt it with new foam and sealed it up nicely, I am worried that connecting the water lines will return me to a heated state. Is there a way to counter act this?

The problem is the way these non-AC boxes work. Water is always going through the core so its always hot. Even with new foam I can feel a breeze at my feet when I am driving because air is being pushed inside the passenger damper and into the box. The air goes in front of the core (not through it) and then passes through the vents until it hits a foam door. Some air leaks of course.

I am guessing that if I open the passenger damper then outside air will be forced to the passenger's feet instead of through the box where it could be heated by the core. I think that should stop air from flowing near the core which should mitigate the oven-state I wish to avoid.

Am I close to the mark? Is this the way to keep the non-AC car cool when the core is connected? Is there a better way that still looks stock?

Thanks again guys!