Remove the spark plug washers or not?

Your '63 didn't come down the line with a '62 head; these references to '62 and '63 are to model years, not calendar years. See this thread. And while you're at it, make sure you're putting in plugs worth having. The NGK № ZFR5N (stock № 3459) is a worthwhile and inexpensive upgrade from standard plugs, see here for details—but you'll need a smaller spark plug wrench (socket) to install these versus the larger socket to remove the standard plugs. Be sure to change your spark plug tube seals while you're in there.