I have a shapefile which contains a lot of polygons which define airspace. These polygons overlap and this is fine (e.g. different airspace configurations for the same airport).
My problem is that many of the polygons could share common vertex but instead they each have a minutely different list of points. This can be seen in the image below: the inner circle is defined twice with points that are almost in exactly the same place but not quite.
What I am trying to do is replace points that are currently extremely close with a single point and use this for all polygons in the layer. It's not realistic to do this manually. I've tried using QGIS dissolve and also eliminate sliver polygons but these seem to be intended for polygons that are not supposed to overlap. It's going to be difficult to convert to lines and back again without losing details associated with overlapping polygons.