I have a 2004 Custom Shop '56 Eric Clapton Blackie. I bought it used. My issue is that my amp plug only fits half way into the output jack on the guitar. It still sounds good, but I'm worried that by not having the plug fully seated will cause issues with the internals, and am wondering if it is a replacement part from a different guitar or if Fender made a "short" jack and if I should look into replacing that jack. Any thoughts?

Half way ? If it's only going half way, then it's not making any contact whatsoever with the jack socket. Can you clarify a bit more ?

Try unscrewing the 2 screws that hold the jackplate on the guitar, then lift it up a little and try plugging your cord in. The wires might be tangled in there, or the plug might be hitting something.

The recess for the jack socket on Clapton strats is a little tight. Reason being that guitar has a stereo socket for the negative side of the 9v battery. Like an effects pedal, the circuit is broken once the jack plug is removed.
So always fully remove your jack plug.

Anyway, because it's so tight in there, wire in that cavity has to be at a minimal amount.
Also they can be stiff as there isn't a lot of room for the tip contact on the socket to flex. So firm but gentle force should do the job. But unscrew the socket plate and have a peek in, make sure the wires aren't tangled across where the jack plug sits.

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No no and no

If I understand your situation correctly, the input jack goes in a bit then gets kinda stuck without locking in place.

Had a similar issue with one of my strats.

Try to unscrew the two screws holding the output jack down, the connector plate (the flat metal strip that's bent at the end) could be positioned incorrectly. It's possible someone wanted to tighten the nut on the outside of the jack and ended up twisting the connector plate in the wrong direction. The strip should positioned away (6 oclock)from the jack piece.

Hope this helps.