If $fg_n$ converges in $L^p$ for all $f\in L^p$, then $g_n$ converges in $L^\infty$.

The claim is not true.

Consider $X=(0,1)$ with the borel measure. and consider the functions $$ g_n(x) = \chi_{(0,1/n)}. $$ Note that $g_n(x)\to 0$ a.e. in $\Omega$.

Then for all $f\in L^p$, it can be shown that we habe $$ \| f g_n \|_{L^p}^p = \int_{(0,1/n)} |f(x)|^p \mathrm dx \to 0, $$ so $f g_n$ converges for all $f\in L^P$ to $0$ in the $L^p$-norm.

However, $g_n$ does not converge to $0$ in $L^\infty$-norm, because $ \|g_n\|_{L^\infty} = 1 $ and using the pointwise convergence, only $g=0$ would be possible as a limit.

Alternatively, it is easy to see that $ \|g_n - g_m\|_{L^\infty} = 1 $ for $n\neq m$, and therefore $\{g_n\}_{n\in\mathbb N}$ is not a Cauchy sequence in $L^\infty$, and thus not convergent.