So the characters are still words, right?

Most likely yes. All characters in Chinese are defined jointly by the way it’s written, the pronunciation, and meaning. You can’t invent new characters like you would a new English word and have something that can be read out loud because there’s no system for deriving pronunciation from the written character itself.

I say most likely because there are still some characters that are phonetic in that their meaning is just the sound, but these don’t cover the whole spectrum of possible sounds in the language as far as I know. They also wouldn’t look as nice in tattoo form since they all use the same radical.