So it's a bit complicated but here is the gist... I made a design using blending modes with a background texture of cardboard. The idea is that the design will get printed directly on the cardboard so now I would need to basically extract those images and get them into a .PNG with transparency. But whenever I do that the white still stays left behind from not having a texture underneath it. I tried using a mixture of masking and blend but the white areas stay behind and it just looks so much worse.
I am not sure you are planning your print the right way.
As I understand your project will be printing black and green on the cardboard:
So you should separate your files, a plate for the black and a plate for the green.
But if you want white over green, you need a white plate that will look simply the same as the black plate, using white opaque ink. There is a chance that you can use a flat green color, without the shape of the pot. Probably not. It depends on the absorption of the cardboard and how opaque the inks are.
You do not need a transparent PNG, no. You need a black and white plate.
Keep in mind that cardboard absorbs ink like water on a desert, so make some tests first.
There is a chance you need to "darken" your white plate.