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I'm not planning on plagiarizing, but I am worried that some of the phrasing I use may have been used before. I'm unfortunately full of cliche's

When uploading ones PS to the LSAC, is there any sort of filter like Turnitin?

LSAC doesn't handle that, different law schools have different ways of screening. Some are using the new "turnitin admit" program, others have their own methods. Don't plagiarize.

The easiest way to do this is to eliminate all cliches. They make your writing pedestrian and dull. Go through your PS, search, and destroy.