General Discussion

Democrat Party, democrat candidate, are all incorrect.

OTOH, language is a living thing & evolves, so eventually dictionaries might accept democrat as an adjective, simply because people use it that way. There was a time when the word despicable had the accent on the first syllable, but after Daffy Duck it's now more accepted to put the accent on the second syllable. That's a matter of pronunciation, though, not usage, so who knows about the word democrat. However, I'll never use democrat as an adjective, even if it becomes acceptable.

If you like the history of language, there's a wonderful book if you can find it, called "There is no Zoo in Zoology" by Charles Harrington Elster.