Democrat vs. Democratic
It isn’t common English usage to use the term Democrat to modify a noun. Your comparisons are odd since Republican was originally an adjective that was then turned into a noun because of its usage in for advocates of different stripes and then the party.
The examples are rather odd, but to give a more comparable example it would be similar to using the despot for despotic. One would not say someone is a despot ruler, one would say they are a despot or a despotic ruler.
Despot ruler makes one sound like a rube as does Democrat Party.
I often find this debate rather bizarre. How many times do we here that young black men should use English properly? I tend to agree with Jesse Jackson that they should, but I also think Republican party members ought to use English properly as well.