I'm looking for an expression that would describe a statement that's too obvious, making it work against its intention.
An example would be to tell someone you didn't steal something, before anyone even asked.
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You have the famous Latin saying:
- an excuse that has not been sought [is] an obvious accusation
More loosely, "he who excuses himself, accuses himself"—an unprovoked excuse is a sign of guilt. In French, qui s'excuse, s'accuse.
The lady doth protest too much, methinks
The quote comes as a response to a question about a character in a play-within-the-play and, as one author wrote,
...the only criticism she has to offer is that the Player Queen's protestations of love and devotion sound too fulsome to be convincing even to the fatuous Player King
So, by protesting (denying or espousing something) too much, one appears to believe the opposite.
"You'll know he's guilty if he says he's innocent before you've even mentioned the crime. That's his tell."
from Google definition
an unconscious action that is thought to betray an attempted deception.
TV Tropes calls this one the Suspiciously Specific Denial.
A False Reassurance works because the speaker is being vague and non-specific enough to pull the wool over someone's eyes. A Suspiciously Specific Denial, on the other hand, fails because the speaker is Saying Too Much. This may be unintentional, such as when the speaker is panicked, is a Bad Liar, or perhaps just a little stupid.
I'm not sure if this would work in the context you have in mind, but how about "the guilty dog barks first"? I think the meaning is basically self-explanatory, and you can see it is common enough to appear in 33 books in this Google search: https://www.google.com/search?tbm=bks&hl=en&q=%22guilty+dog+barks+first%22