See Mastering the Spelling Checker.

AFAIK, OKing a word's spelling on one computer does not carry into next computer to use it.

Click on one of the misspelled words that isn't being flagged. On the Review tab, under Language click on "Set Proofing Language." Are either of the checkboxes at the bottom checked; neither should be. If the one for "Do not check spelling or grammar" is checked, don't bother to uncheck it now. Instead get out of that dialog and back to your document. Select the entire document and check the status of that box again. Make sure it is clear, not checked or colored-in.