In American English, the phrase "the lady doth protest too much" definitely indicates that you think the subject's ardent denial of a proposition is meant to cover up its embarrassing truth. For example, if you offered someone a beer and they said "No thanks. I absolutely hate beer. Never touch the stuff, ever. I can't even imagine ever wanting to drink a single drop of that filthy, mind-numbling liquid. It's poison to me. I absolutely do not want a beer," you might well think that they were trying hard to restrain themselves from taking it.

In Hamlet, queen Gertrude is watching a play in which a faithful widow refuses to marry too soon after her husband's death. Hamlet arranged this as a veiled accusation, because she married his paternal uncle just after his father died... so quickly that, as Hamlet quipped, the food at the wedding banquet could have been leftovers from when they had the funeral. Gertrude says that the widow is protesting her suitor's advances too much, evidently because she's ashamed that she failed to do so.

If we take Gertrude at face value, this comment means just the opposite of the current form. She says it not because she disbelieves the widow, but because the widow's sincere protests make Gertrude feel inferior. One way to reconcile things is to assume that Gertrude does not believe the widow, and thinks that the widow, like her, really does want to marry. Then her comment means what we mean by it today, that the subject is being hypocritical.