In every case of $\neg \neg P$ that I've come across, the statement $\neg P$ has been disproven. Never has such a proof merely been proof for the inexistence of a proof for $\neg P$.
Take the proposition $\neg P = 3/2 \in \Bbb Z$. $ P = 3/2 \not \in \Bbb Z$. I am now going to disprove $\neg P$.
$3/2 = 1.5$.
I have now just proven that $\neg P$ is false, or in classical terms, $\neg \neg P$. It is thus simple logic that $P$ is true, because I didn't just prove the lack of a proof for $\neg P$*, I proved directly that $\neg P$ is false.
*I think I've read that ZFC has yet to be proven consistent, so if I'm doing this in ZFC, then I haven't even proven that there is no proof for $\neg P$; yet, my proof for the falsity of $\neg P$ is still undeniable.
If I'm not mistaken, intuitionists agree that if I prove $\neg P$ false, then I automatically prove $P$ true. They just disagree on what a mathematical disproving of any/specific negative statement(s) are/is doing. They seem to think that disproving a negative statement isn't proving its falsity, just its lack of a proof. I just don't understand why they think that. As far as I understand, it must be something that only applies to specific negative statements.
Intuitionists agree that if you prove $\neg Q$, you are proving that $Q$ is false. For an elaboration on this idea, see my answer here. This is true in the special case of $Q = \neg P$.
You are mistaken. This is exactly the thing intuitionists do not accept about classical logic.
As for your purported proof of $\neg \neg (3$ is odd$)$, you would need to seriously flesh out this proof (starting with a definition of “odd”) before I can say anything about it.