Audio & Visual recording solutions - help much appreciated

Hello everyone, I'm hoping someone could help me with a project I'm working on.

Here's my goal :

1) To record a live DJ'ing set using video coming from my camera and direct audio coming from the mixer.
2) To simultaneously Livestream while recording.
3) (BONUS) have the ability to swap between multiple feeding camera's.

I recently attempted to record this with Quicktime on my mac and ran into a lot of trouble with blanking and crashing over a long period of time. Furthermore I cannot swap between camera's using quicktime.

If anyone know's the best software and/or method to do this I would be extremely greatful.

Thanks!

Liam