I'd like to move the character's words to a new line for a dramatic effect (I can do that right?). But is a colon appropriate here? or Or should I just stick to
It is correct the colon is out-of-date, but take note of your last two examples: He shouted defiantly. "I won't give up!" In this there are two separate thoughts. One, the man shouts. Two, he says, "I won't give up!" I don't think this is what you're intending He shouted defiantly, "I won't give up!" In this you're describing the way he is saying the phrase. Some might say over describing. It is obvious from his word choice that he is defiant. You might try: He shouted, "I won't give up!", or just "I won't give up!" by itself.
Avoid, but don't necessarily eliminate. I've just finished reading a book that was long-listed for this year's Man-Booker prize, and the lack of adverbs of manner started to feel odd after a while even though they hadn't been eliminated completely (there was one about every two or three pages). By comparison, this year's Philip K Dick award-winning trilogy has a few per page (lower when it's mainly dialogue). The objective is to make it good to read, not as short as possible. The resistance to adverbs of manner comes from those novices who use a few per sentence. I don't think it's as redundant as StarChild suggests. If you drop the "shouted" then he might be muttering it under his breath. Even with the "shouted" I can hear a difference between it being shouted defiantly and being shouted angrily. That's not to say that just writing "I won't give up" wouldn't be better, it's just that it's not a simple choice. The good writer will be conscious of a whole spread of possibilities, will be aware of the different effects of those possibilities on different readerships, and will choose the one that is best for their particular purpose.