I have a polygon (A) that I drew the eastern border of in QGIS. I have another polygon(B) whose western border I want to nest against the eastern border of polygon a so they are touching but not overlapping each other. Is there away I can do this other than reshaping polygon B's western border by hand as I had to do with polygon A's eastern border?
Go Settings -> Snapping options and enable "Avoid intersection" For your layer. Then just overlap borders of the second polygon over the first polygon and the second polygon will be cropped. See pictures.