Just my 5 cents. A used radiator is always a bit of a crap shoot. Radiators that are now 60 years old? If you replace the mounts I would spring for the new. Now to the mounts. The rear mounts are difficult to access in truck but can be done. Not unusual for both front and rear to have lots of movement if original. The rear mount has one bolt through the frame and two into the bell housing. The one on the frame can wallow out and is most likely loose, I have also seen threads ruined by bolt movement at the housing. Most likely this frame bolt is loose. The front mounts can also be loose but are easier to change. There are threads here on that subject. The tub and top plate are not reproduced so save them. Buy a new shield and rubber insert and new bolts where needed for this. You will need to raise the motor a bit for clearance. The time to do it is now with the radiator out.
New rear mounts are available from most vendors, they are rubber covered steel, there is also a HD mount in steel available NOS only on eBay infrequently. Bottom line is I would change both of the rears and rebuild the front before putting in a new radiator. Denny Graham has some good pictures of the front mount on his site I have pictures of them to in my reference folder. Good Luck!

1950 Chevrolet 3/4-Ton 3600
More pictures on Flickr