This is a very common engine and transmission. The engine "350" was not an orignial for 1963 but the same basic block and probably bolted right in.
The tune up items need to be checked and the throttle linkage must be set such that when your foot is on the floor the carb is wide open at the same time.
That doesn't account for the shaking. That could be plugs, wires, distributor, exhaust problems and on.
Saw this posted in general discussions also and it's best to just post it once. I'll check there too and see if you have any replies and can move the thread but that won't help now,
Are you doing some or all of the work on this project? It's hard to not be VERY involved with an old car and know all your parts sources. just about anything you could want is available for this car without too much trouble,
T
Tom Greenleaf - MetroWest Boston - USA
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http://www.autoacsystems.com/tomgreenleaf/