I define the following wheighted Sobolev spaces $$L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$$ and $$H^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg| D^\alpha u\in L^{2,s}(\mathbb{R}^3),|\alpha|\leq 2\bigg\rbrace$$ I know that the classical Sobolev space $H^2(\mathbb{R}^3)$ is contained in $C(\mathbb{R}^3)$ because $2>\frac{3}{2}$. Can I extend this result to the above weighted Sobolev spaces?

Continuity is a local property: functions which are in your $L^{2,s}$ are locally in $L^2$, so functions in $H^{2,s}$ are locally in the Sobolev space $H^2(\mathbb R^3)$, thus are continuous functions (even Hölder 1/2-$\epsilon$).