Angela, welcome to the best British Car site in the world.
Incidentally, I have a '63 as well.
The occasional squeaking noise you hear bothers me a bit. I'd pull the drums and make sure the shoes are properly assembled first. If all is OK, try this:
Make sure that the rears are properly adjusted. In your case, I would disconnect the emergency brake cable for now, and adjust each side until you can't rotate the wheel by hand. Then, back off the adjuster until the wheel will <u>just</u> spin without the shoes dragging.
While the cable is disconnected, make sure it moves freely in its housing, and check to make sure all of the linkage and pushrods are free to operate. A small dab of white grease at the pivot points won't hurt.
There is a grease fitting on the cable that you should hit with a few pumps of a grease gun. If everything looks good, reconnect the cable and proceed.
To adjust the emergency brake, pull the lever to the third click, and adjust the cable using the nut at the rear of the cable housing until you can just barely rotate the wheels. Release the brake, and make sure the wheels now turn easily, with no drag.
Try this, and let us know how it turns out. We may have to get deeper into the system, but hopefully this'll take care of the problem.
Dragging rear brakes are no fun, and they can cook axle seals in short order.
If you don't yet have a manual for the car, I'd suggest the Bentley manual for the 948-1098 cars.
Good luck,
Jeff