A term for someone denying an accusation but appearing guilty as a result

In this example, you might be protesting too much.

Shakespeare introduced the phrase in Hamlet: "the lady doth protest too much, methinks".

"The lady doth protest too much, methinks" is a quotation from the 1602 play Hamlet by William Shakespeare. It has been used as a figure of speech, in various phrasings, to indicate that a person's overly frequent or vehement attempts to convince others of something have ironically helped to convince others that the opposite is true, by making the person look insincere and defensive. _(Wikipedia)

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The phrase can mean the denial/protest is too frequently repeated but it can also just be due to the vehemence or phrasing of the denial. It can be thought of — as the Wikipedia article on the line in Hamlet says — a subtle, unintentional apophasis.

Apophasis is a rhetorical device wherein the speaker or writer brings up a subject by either denying it, or denying that it should be brought up. _(Wikipedia)

A single misplaced word can cause others to think the person protests too much.

Accusing someone of protesting too much is a not-so-subtle form of double bind. To deny it is to protest even more.

We often say, "He's digging his own grave," when someone's attempt to defend themselves makes them sound guilty. (US)

Hmm, I can't quite recall something that elegantly covers this exact case. Here are lots of pieces to work with though.

Doing anything that ultimately damages yourself can be described as, "digging your own grave".

Specifically saying something that is contrary to your best interest is, "putting your foot in your mouth".

The act of doing anything that incriminates yourself is just self-incrimination.

Jon could have stopped talking, but he self-incriminatingly put his foot in his mouth instead.

Answering the question/accusation: “You killed him, didn’t you?/Why did you kill him?” with anything but a firm denial gives the impression that one is dodging the question and therefore can raise suspicion that the truth is being withheld. (Wikipedia)

Asking rhetorical questions in response to questions certainly has its place and is not always evidence of ‘question dodging,” but even if one fancies themselves as possessing the intelligence and logic of Socrates, responding to a murder accusation by asking such a question will, at best, not result in being excused from further interrogation and at worst is tantamount to “shooting oneself in the foot.” (without getting into the cited ELU discussion concerning whether that phrase means ‘intentional,’ ‘unintentional,’ or either kind of self-harm)

Perhaps it's not an actual term that people use, but perhaps we should reference it as an "appeal." Maybe "Appeal to Motive," or something along those lines. And then we all can just agree that "Appeal to Motive" is a blanket statement for defending or attacking someone's views/actions based on establishing a motive, or lack thereof, to think/do that thing.

the phrase (as) guilty as sin Merriman-Webster

completely, or very[sic] guilty

As in:

Despite his protestations, he appeared in body and manner guilty as sin!