Do I need to check the words AND the image in my design?

Yes — they're two separate risks, and clearing one says nothing about the other. Word marks protect phrases and names ('Just Do It'), while design marks protect the visual element (the swoosh). A phrase checker can pass your text while your graphic still matches a registered design mark — and a clean graphic doesn't clear a trademarked slogan in your title or artwork. Most takedown-triggering conflicts on POD platforms are visual, which is the side text-only tools can't see: LogoVerify reads your design's image and any text in it against 13M+ USPTO marks in one check. (General information, not legal advice.)