Neighbour refuses to agree to newly built wall?

What I'm hearing is that your neighbors don't want you to do anything at the place your two walls join. OK, its not symmetrical - but you could do the "saw the baby" in half thing, and NOT tie your wall to theirs in any manner or shape so as to provide strength to the whole. You don't mention how tall your wall is - if its below 4 feet you can get a way with a 3 inch gap between your wall and neighbors, if 6 feet tall wall - 2 inch gap. It's not pretty, but it gives you the option to change the style or height of your wall.