Was Shakespeare's Hamlet truly fat?
Good question. The OED doesn't sort that cite into any of its senses, but none of them are remotely "rusty". The original and base sense is fatted, fat used for an animal that is full of fat and ready for slaughter.
By extension for people, variously:
2.a. plump, having fat. b. obese, overly fatted.
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4.c. juicy [in reference to actor's roles, but not attested til 18th c.]
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9.d. loaded [in reference to a prisoner's ability to get a good ransom]
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10.a. rich
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- slow, satisfied, dimwitted [as a fatted animal]
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- plump, rich, &/or dumb in combination
It looks like your Shakespearean scholars are trying to twist sense 11 into something nicer than it is.
At best, Hamlet is being called 'porky' (Sparknotes 'modernizes' it as 'flabby')—which matches his difficulty breathing and I seem to remember fits references elsewhere to his interest in food, although I may be confusing him with Jesus in that regard. At worst, he's literally being called porky—his intellect and movements bestially sluggish and suggestive of an animal ready for slaughter.
At the same time, it's not gospel. She's responding with motherly worry to Claudius's blithe comments about how well Hamlet is doing in a deathmatch. (He's talking up Hamlet since he's about to poison him even if the kid is lucky enough to beat Laërtes.)
If you were really curious... [The Slate article is actually very thorough, even if it does go on too long about sweaty before admitting that sense is not accurate at all.]