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For my diversity statement, I end with " I hope to blah blah at XYZ law school." Is this bad that law schools know I essentially changed the last name for each statement? They all ask you to specifically address what you can do for their university, so I feel like it should be acceptable. I still feel apprehensive about it...

I think it's the sort of thing everyone tells you not to do but they secretly do anyway. It's actually far easier to just remove the name and come up with a better conclusion for your essay and just send the same essay to all schools instead of name-dropping and changing it for every school.

That being said I don't think it's a major issue if you did it.

tl;dr - get rid of it and conclude the essay better, if not don't stress out too much about it.