‘64 Dart Carburetor Help

I’ve got a 1964 Dodge Dart with a 270 Slant Six engine. I believe it’s 225 CI displacement.
I adjusted the carburetor’s idle to the best of my ability, and yet it still idles very fast. It’s causing hard shifting, to the point that my tires spin a bit when I shift into reverse from neutral.

My engineering teacher took a look and adjusted it, but he thinks I’m missing a gasket underneath the carburetor, and that a piece on the throttle linkage is broken.

I can post pictures if needed, but I’d really appreciate some advice, as I don’t even know where to start.