Paz and Zurek (in ISBN 3-540-43367-8, p.91) give the following calculation:
For $|E_1(t)\rangle= \bigotimes_{k=1}^N\Bigl(\alpha_{k}e^{ig_kt}|{0}\rangle+\beta_ke^{-ig_kt}|{1}\rangle\Bigr)=|E_2(-t)\rangle$ the inner product of $|E_1(t)\rangle$ and $|E_2(t)\rangle$ is
$\langle E_1(t)|E_2(t)\rangle=\prod_{k=1}^N\Bigl[\cos{\left(2g_kt\right)}+i\left(|\alpha_{k}|^2-|\beta_k|^2\right)\sin{\left(2g_kt\right)}\Bigr]$.
Is there a mistake? My calculation led to
$\langle E_1(t)|E_2(t)\rangle=\prod_{k=1}^N\Bigl[\cos{\left(2g_kt\right)}+i\left(-|\alpha_{k}|^2+|\beta_k|^2\right)\sin{\left(2g_kt\right)}\Bigr]$.
I get the same result as you.
My attempt:
We can reduce to the case $n = 1$ to get rid of the tensor product. So we need to compute $\langle E_1(t)|E_2(t)\rangle$ where
$|E_1(t) \rangle = \alpha e^{igt} |0\rangle + \beta e^{-igt} |1\rangle \implies \langle E_1(t) | = \bar{\alpha} e^{-igt} \langle 0 | + \bar{\beta} e^{igt} \langle 1|$ (assuming that $g$ is a real constant, and that $\alpha$ and $\beta$ are arbitrary complex numbers). And $|E_2(t)\rangle = |E_1(-t)\rangle = \alpha e^{-igt}|0\rangle + \beta e^{igt} |1\rangle$.
So $\langle E_1(t)|E_2(t)\rangle = \alpha\bar{\alpha} e^{-2igt} + \beta\bar{\beta} e^{2igt} = |\alpha|^2(\cos(2gt) - i\sin(2gt)) + |\beta|^2(\cos(2gt) + i\sin(2gt)) = (|\alpha|^2 + |\beta|^2)\cos(2gt) + i(|\beta|^2 - |\alpha|^2)\sin(2gt)$.
Which is the same as what you got if we assume $|\alpha|^2 + |\beta|^2 = 1$. I haven't had a look at the original paper, but from your post, I think your calculation is correct.