Instead of using Multizone try with the Sweep method and see if that helps.
and As peteroznewman mentioned, we are not allowed to download files. so can't work on your actual model, but can only give suggestions.
To get rid of the worst quality elements in the bottom part, you will need to either refine the mesh a lot or get rid of the almost tangency as Peter has mentioned. For the OGrid that you are cutting, I would use it as a last resort if default mesh with the sweep is too bad. I would first check the strategy that I mentioned in my first response. You can cut the bodies using cylindrical faces and arbitrary planes, right? Note I am not asking to split face using lines, I am asking to split bodies using surfaces.