Sign languages absolutely have words, they're just formed out of finger and/or hand positions + finger and/or hand movements, and sometimes facial or other body

Sign languages absolutely have words, they're just formed out of finger and/or hand positions + finger and/or hand movements, and sometimes facial or other body movements, rather than out of sounds (phonemes). And yes, these signed words are specific to each sign language (although there is borrowing between many sign languages, just as there is among spoken languages). Sign languages also have grammars, as do spoken languages, although the grammars of sign languages may allow for some things to be encoded simultaneously rather than sequentially, as is usually required with spoken languages.