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I have searched for the audio recorded on the CVR from the US Airways Flight 1549, but all clips and audios I found seems to be audio recorded from the flight control and not from microphones inside the cockpit.

For example:

In this clip, we only hear the pilot sparsely, but the air traffic control and the airport tower controls quite often. Although, we can also see a transcript of what is said in the cockpit, so there obviously exists some recordings from the cockpit.

Where can I listen to the audio recorded from the cockpit and not from the air traffic control?

You can't, because it is in fact forbidden to release CVR audio under US law. From the NTSB:

The CVR recordings are treated differently than the other factual information obtained in an accident investigation. Due to the highly sensitive nature of the verbal communications inside the cockpit, Congress has required that the Safety Board not release any part of a CVR audio recording.

Other countries may have different rules.

This rule came about after the crash of Delta Air Lines flight 1141, (commented by @MichaelK)

[...] where the crew — in extremely cruel irony — were bantering on the flight deck about what to say that would get captured on the CVR in case of a crash. That irony was then squared in that the bantering was a directly contributing cause of the crash, in that they forgot to deploy flaps and slats

You can't listen to it due to the regulation mentioned in Sanchises' answer, but you can find the transcripts. There is one here, for example.