In episode #243 ("The Moon is Not Blue"), the camp watches the movie "State Fair", which was released in 1962!

I don't recall that episode specifically, but were they really watching the remake with Pat Boone and Ann-Margret? The original musical version was released in 1945 (and a non-musical version with Will Rogers and Janet Gaynor was made in 1933).

I do know that the series was awash in anachronisms (aluminum beer cans, films and books not yet in existence, etc.), but that might not have been one of them.

no, that wasn't a mistake. They WERE watching the 1945 version.