Not a complete solution, but a strategy showing that $\liminf F(n,n)>0.4$.

Player A will pick some positive constant $c$ and accept the first draw greater than $1-\frac{c}{n}$. The probability A sees such a draw is asymptotic to $1-e^{-c}$, and the value of the first such draw is uniform in $[1-c/n,1]$. The probability that A wins using this strategy is asymptotic to
$$
\left(1-e^{-c}\right)\cdot\frac{n}{c}\int_{1-c/n}^1 t^n\,dt\to \frac{\left(1-e^{-c}\right)^2}{c}.
$$
This is maximized by taking $c\approx 1.25643$, giving a probability $\approx 0.407264$.