Is sounding natural really so important

I've been studying Japanese for a few years and can communicate quite well but I don't sound natural. I kinda sound like a news anchor all the time. I always use polite words and make long full sentences. I can't really speak the way native speakers do. In chunks of short sentence fragments with dropped particles and omitted parts of speech.

I'm a little worried about that. What do you think?

Another thing I think about are mannerisms and habits. I don't 'act' Japanese because I am not. However, I find that foreigners who do act Japanese are seem to be more accepted.

In everyday life, being able to get the message across is much more important than sounding natural. After living in Japan for a few months, you will quickly become more natural sounding.

If "acting non-Japanese" means violating local manners, it is, of course, not going to help you. I recommend to get familiar with the rules of etiquette, while not getting too conscious and nervous about them, at the same time.

When foreigners arrive in Japan, they will always stick out due to language and culture differences. But with an open mind and a friendly attitude, you can avoid frictions and will gradually be able to adapt.

Can you give some examples of behaviour that you would consider to be "acting Japanese" or "not acting Japanese"?

As for sounding natural, that's unlikely to happen in a few months. If you are here long enough though, and make an effort to absorb the intonation and speech patterns you hear, then over time you will begin to sound more natural yourself.

Not sure if you are living in Japan or not, but sounding more natural comes mostly from absorption. When living in any environment, you tend to act and do things like the majority of people in that specific environment.
As for language, there are no better teachers than the natives themselves so it is good to listen to them a lot and learn how they speak in regards to words, context, grammar, etc. If you are aware of it when speaking youself you will gradually start speaking more 'natural'.
As for actions, well, that kind of depends who you hang out with. Also, there are probably many actions, etc. that you will not do simply because you have not been brought up in the same environment.