You did the right thing, if all products in the cart have "Requires Shipping: No" then no shipping options should show up. I think the problem is probably checking out with Amazon -- if it's not coded to detect the products without shipping, then it may need to be modified to do that. Try using the normal "Checkout" button instead, and see if the shipping steps of checkout show up. If they don't, they you'll know it's working right, and it's just the Amazon Pay extension.
I'm not sure if there's a free fix available, but most likely not. If you need to find a developer to hire to fix it, you can post a request in the OpenCart
"Commercial Support" forum, which is checked by a number of OpenCart developers. You can also try checking out the OpenCart
"Partners" area.