Try this with ovals. I'll use the colors in the sample that you shared.
First thing, mark the top of each layer (cut to final size for your layers--4" x 5-1/4 in your sample) lightly in pencil on the back of the paper. This is so that you keep the alignment uniform. Match the pencil marks in every step below.
Place your smallest oval die on the green paper. Make sure it is exactly where you want it to be. Tape in place.
Put the green layer on top of the pink layer and tape them together. Run both through your machine together. The green layer will cut and the pink layer will impress. Untape all of it and take apart.
Find the next larger die that you want for the pink layer and place it evenly around that impressed mark from the smaller die. You should be able to see it easily, and you'll be putting it pretty close to that impressed mark. It's easier to see that they will be lined up that way. Tape in place. Now put the cream one behind the pink one and tape them together, then run thru your machine. Take them apart and you have the green and pink perfectly aligned.
Now you take the next larger die and place it around the impressed mark on the cream piece and tape, then cut.
Your three layers (remember all the marks are on the back at the top) are perfectly aligned. When you layer with dimension, you'll make sure that the outside edges line up. Then the ovals will stack properly.
Hope that's understandable!