You need to remove the rear brake drums and see what's what .
Being a '51 it should have the vastly improved BENDIX brakes that parts are readily available for and dirt cheap as these brakes were used up through the 1967 model year .
Quite often there's simply decades of accumulated oil and brake dust on the shoes and a good cleaning will set things right again .
Remember to never,
EVER turn or machine brake drums ! .
I use the SAM'S CLUB commercial degreaser, it comes in gallon jugs and I cut it 1 part degreaser to three parts water, it's amazing stuff, you'll need some bushes including soft bristle paint brushes to get through all the crud, don't forget the wash the drum inside and out .
When the drums are off is the proper time to remove, clean and lightly grease (using white Lithim based grease) the threaded adjusters .
Post up some pictures of your new truck, we don't mind 'em dirty or rusty =
.
