OK ... from everything you've said, it definitely sounds like it's the quill itself that's moving. The DRO evidence is a real convincer.
I looked at the mill on Grizzly's website and checked through the manual. It looks like the quill locking lever (236) is a bit flimsy - can't tell from the diagram, but it looks like it might be M8 or something about that size. Then there's the brass pin (237) on the end. Possibly inserted into a hole in the end of the screw. I'd try removing the lever and pin, and checking to see if the pin is loose in the screw. It might also be the case that the pin is a bit too long for the hole in the screw. If so, the pin could pivot at its "inner" end, allowing the quill to move up and down.
If the quill lock is indeed the problem, a fix might involve shortening the pin's shank slightly, re-making the pin so it fits more closely in the screw, or maybe even drilling and tapping the mill head for a beefier screw.
PS - I gotta politely disagree with RJ regarding mill pullout, at least in this case. If you were side milling or taking a much deeper cut, mill pullout would definitely be a consideration. But if you're having a problem at 0.005" or 0.010" DOC, the downward force on the mill would be miniscule.
Thanks Hman! I may remove the lever (from 236) and wrench tighten with more torque to see if that makes a difference. I've contacted Grizzly Customer Service and am eager to see what they say after they look into it. The machine only has a couple hours of use so I'm a little concerned that this is an issue already. I really appreciate you taking the time to look into this!