Jack up each corner of your vehicle just enough for the tire to clear the ground. Spin the tire and see if it catches the ground or you notice an out of round condition.
You might have cord separation in the casing producing an "egg" on the tread.
Less likely on a new tire but not unheard of either. Could cause a blowout.
If this condition exists you should get the tire replaced right away.

 

it may very well be tires. but it would have to be the rear tires as you stated that the vibration is through the body. bad front tires would transmit through the steering wheel.


ujoint will transmit through the body. the only way to check for tight ujoints is to remove the driveshaft and move the joint by hand; to feel for roughness. the ujoint will still be moving when you did your test, shifting into neutral; that test will not rule them out.

 

If its an automatic and you turn off the engine the transmission fluid stops also. Its what lubricates and cools your transmission. At 60 mph it wont take long to do damage.