Welcome to the forum. I, and I'm sure others here, would like to help, but I'm confused. Maybe if we clear up a few things it will help. First, you said "I have a "Part" file that contains numerous 2D Drawing sheets." SolidWorks has three basic file types: Parts, Assemblies, and Drawings. Part files can't contain Drawings. Maybe you mean you have a multi-body Part, and a Drawing with a separate sheet for each body? Or did you mean you have numerous sketches that you're trying to copy to another Part file?
Also, I'm struggling to figure out what "I am in the process of saving all of these sheets individually, each into a new "part" file, but whenever I try to copy + paste them into the documents, a message pops up saying that the text "cannot be copied to the clipboard" (Such as part labels/bill of materials labels)." means. Again, you can't save drawing sheets into a Part file. You can copy drawing sheets from one Drawing and paste them into another Drawing, but that shouldn't result in the error message you're seeing.