The usual way of saying this would be "I feel (very) (shocked and) ashamed".
I feel very shocked and shame. -- for this sentence to work, "shame" would have to be an adjective, which it isn't.
I feel much shocked and ashame. -- I was about to assert that there is no such word as "ashame", but according to one or two dictionaries there actually is. It apparently is a verb meaning "to shame". This is news to me; it seems to be obsolete and/or very rare (except as a mistake for "ashamed" or "a shame", of course, which I assume accounts for the bulk of the 467,000 hits that Google reports). Anyway, it doesn't work in this sentence. "much shocked" is not wrong, but it sounds rather formal and literary. In everyday contexts you would usually say "very shocked".
I feel shame. -- OK, but "I feel ashamed" is more usual.