Your conjecture is true. First fix a $\Sigma^1_{2n+1}$ formula with a natural number variable $n$, which we can write in the form $\exists x\in \mathbb R\ \psi(x,n)$ with $\psi(x,n)$ in $\Pi^1_{2n}$. For any $n\in \mathbb N$, $(\exists x\in \mathbb R\ \psi(x,n))^{M_{2n-1}}$ holds if and only if $\exists x\in (\mathbb R\cap M_{2n-1})\ \psi(x,n)^{M_{2n-1}}$ holds, or equivalently $\exists x\in Q_{2n+1}\ \psi(x,n)$. In the last step, we use that $\mathbb R\cap M_{2n-1} = Q_{2n+1}$ and that $M_{2n-1}$ is $\Pi^1_{2n}$-correct. Since $Q_{2n+1}$ is $\Pi^1_{2n+1}$-bounded, $\exists x\in Q_{2n+1}\ \psi(x,n)$ is equivalent to $\Pi^1_{2n+1}$ formula. This shows  $(\Sigma^1_{2n+1})^{M_{2n-1}}\subseteq \Pi^1_{2n+1}$.

Conversely, every $\Pi^1_{2n+1}$ formula with a natural number variable $n$ is equivalent to a formula of the form $\exists x\in Q_{2n+1}\ \psi(x,n)$ where $\psi(x,n)$ is $\Pi^1_{2n}$. This follows from the proof of the usual Spector-Gandy Theorem for odd levels $\Pi^1_k$ (see Moschovakis's *Descriptive Set Theory*, 6E.7), which actually shows that for any $\Pi^1_k$-bounded set of reals $A$ that contains all $\Delta^1_k$ reals, every $\Pi^1_k$ formula $\varphi(n)$ is equivalent to a formula of the form $\exists x\in A\ \psi(x,n)$ where $\psi(x,n)$ is $\Pi^1_{2n}$. But we've seen that $\exists x\in Q_{2n+1}\ \psi(x,n)$ is just $(\exists x\in \mathbb R\psi(x,n))^{M_{2n-1}}$, which is $\Sigma^1_{2n+1}$ in $M_{2n-1}$. Thus $\Pi^1_{2n+1}\subseteq (\Sigma^1_{2n+1})^{M_{2n-1}}$.

You might also want to look at Theorem 4.12 of John Steel's paper *Projectively Well-Ordered Inner Models*. I think you can use the proof to get a more inner model theoretic proof of your conjecture, but Steel's result is closely related and of independent interest.