We just replaced the flush mechanism in our toilet- it is the one on the Left in this picture
https://www.askmehelpdesk.com/attachments/plumbing/38543d1324685176-toilet-tank-fills-up-too-much-water-both-ballcocks.jpg

The water keeps overflowing past the max and overflowing out of the hole where the flusher handle goes in the tank. I have tried readjusting the floater but the same thing happened. In another post I saw a response saying that this problem is due to the seal in the fill valve needing to be replaced... but the whole mechanism is brand new. Is there any other option for what could be wrong?

You have a FluidMaster Ballcock. Are you saying that when you lift the float can that the water doesn't shut off? Let me know, Tom

Yeah even when I lift the float the water doesn't stop- it keeps making this noise toilet noise - YouTube (http://www.youtube.com/watch?v=C8Fs7jVoRb0&feature=youtu.be)

I just realized that the valve attaching the toilet to the water line is leaking and needs to be replaced. Could that be related or is it two separate issues?

Let's replace the fill valve and see. Good luck, Tom

You said you replace the flush valve. Do you mean you replace the fill valve?
Is the float touching the back of the toilet. Float does not appear to be moving as toilet fills.
Leak is separate issue.