Just use the fact that
$$ \sigma_X W_t^1 + \sigma_Y W_t^2 = \sqrt{ \sigma_X^2 + \sigma_Y^2 + 2\rho\sigma_X\sigma_Y } W_t $$
holds in probability assuming that $W_t^1$ and $W_t^2$ are 2 correlated Brownian motions with
$$ d\langle W_t^1, W_t^2 \rangle_t = \rho dt $$
and $W_t$ is a new standard Brownian motion defined over the same probability space.
Simply put, just replace your sum of two correlated Gaussians (LHS above) by a single Gaussian (RHS above) exhibitting the exact same statistical properties (for a Gaussian identical mean/variance is enough). Doing so, you can now use the formulas you are accustomed to.
Applying this shows
$$ X_t Y_t = X_0Y_0 \exp((a+b-\frac{1}{2}\sigma_X^2-\frac{1}{2}\sigma_Y^2)t +\sqrt{ \sigma_X^2 + \sigma_Y^2 + 2\rho\sigma_X\sigma_Y } W_t) $$
is lognormally distributed with mean
$$ \mu = \ln(X_0Y_0)+(a+b-\frac{1}{2}\sigma_X^2-\frac{1}{2}\sigma_Y^2)t $$
and variance
$$ \sigma^2 = (\sigma_X^2 + \sigma_Y^2 + 2\rho\sigma_X\sigma_Y)t $$
hence applying the usual formula for the mean of a lognormally distributed variable
$$ E[X_t Y_t] = e^{\mu + \sigma^2/2} $$
is a function of $\rho$.
[Edit]
I think you are completely mixing up two different problems (here two different probability measures).
I understand from your comment that you define the dynamics of the FOR/DOM instantaneous exchange rate $X_t$ (i.e. $X_t = x$ meaning that, at time $t$, 1 unit of foreign currency = x units of domestic currency) under the foreign risk-neutral measure $\mathbb{Q}^f$ (or rather the probability space $(\Omega,\mathcal{F},\mathbb{Q}^f)$ along with the price process of an equity underlying denominated in the foreign currency.
In that case, under the domestic risk-neutral measure $\mathbb{Q}^d$ (or rather the probability space $(\Omega,\mathcal{F},\mathbb{Q}^d)$), one should have, by absence of arbitrage opportunities and assuming market completeness:
$$ \frac{Y_t X_t}{B_t^d} \text{ is a } \mathbb{Q}^d \text{- martingale} $$
with $B^d_t$ representing the time-$t$ value of a risk-free money market account in the domestic economy in which 1 unit of currency has been invested at $t=0$. Using the martingale property it then entails that:
$$ \frac{Y_0 X_0}{B_0^d} = Y_0 X_0 = E^{\mathbb{Q}^d} \left[ \frac{Y_t X_t}{B_t^d} \vert \mathcal{F}_0 \right] $$
and further assuming deterministic rates:
$$ E ^{\mathbb{Q}^d} \left[ X_t Y_t \vert \mathcal{F}_0 \right] = e^{-r_d t} X_0 Y_0 $$
which is independent of $\rho$.
This is known as a compo adjustment in deriatives lingo.
That being said, under the measure $\mathbb{Q}^f$ where you have originally defined $X_t$ and $Y_t$, the expectation $E^{\mathbb{Q}^f} [ Y_t X_t ]$ does dependent on $\rho$ as hinted above.
More info on the quanto/compo change of measure technique here
