I think your rotor is fine.
New rotors come with cross-hatching on their surface.
I think you should take your car for a test drive and test the brakes, then take it on the highway to see if that shake is gone.
Then install your rear rotors and pads, which I assume you haven't done yet?
These are my rear rotors on my DD.
They look like crap and are full of grooves but they work and don't drag and I don't consider them unsafe in any way. They're only a couple of years old.
I certainly wouldn't want to track my car in the condition that it's in though.
I drive the car very gently but I'm sure my brakes would grab hard enough to kick in the ABS if I ever had to do that.