Thanks you for the replies.
I opened up the worm gear housing and that seems to be where most of the backlash is. I tried loosening the housing, but that does not seem to solve the problem. I think the only solution is to completely rebuild the Z from the ground up, scraping the the rack and pinion and worm gear system. A worm gear driving a rack and pinion is not a very good design and just creates mega backlash.
Here are some pics so you can see what the machine design is.
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