https://i.pinimg.com/236x/b0/a9/81/b...ery-fabric.jpg

its kind of like this. On the top side of the fabric the pattern is raised, but it's smooth on the backside. I'm just trying to make sure I sew it to where it lays the same way. And I'm trying to make sure when it is sewed into one long strip, the corners will lay correctly where the fabric is joined.

Basically just use the 'raised side' as the right side and the smooth side as the wrong side, putting it together just like doing regular quilting cotton fabric. To get the corners to lay exactly where you seam/join it will take some doing with measurements, plus -- because it will be a mitered corner since you are doing it all in one long strip, it will be really bulky at the corners to have the joins meet there as well.

I would just join into a long strip and let the joins/seams lay where they fall. With a binding (narrow) it's not something that will stick out once it is all together.

Maybe this video will help?
It is 2 colored binding, which may be similar to your 2 sided binding issue.

https://www.youtube.com/watch?v=EPJW4pYSE-Y

I don't think the angle of your joins will matter in the end....

I have the top and right strip figured out, but that's only the binding for those two sides. Do I just sew the bottom and left together the same way then attach the entire thing? Or do I need to just do it in two sections, then tuck and sew them?

Are you talking about the binding that finishes the edges of the quilt after it's quilted or are you talking about the borders that go around the blocks of the top? Bindings are put on in one long piece. Borders are put on individually by side.

Why not use a cotton fabric for the binding? I used cotton fabric to bind a cuddle fabric quilt and it looked great and much easier. It looked a little classier too.

I keep waiting for an answer to this question.

I do, too.