Suppose that $(X, \|\cdot\|_X)$, $(Y, \|\cdot\|_Y)$ are two Banach spaces such that $X\subset Y$ and $\|x\|_Y\leq \|x\|_X$ for all $x\in X$ and $X$ is dense in $(Y, \|\cdot\|_Y)$.

Every functional $y^*\in Y^*$ also yields a functional $y^*|_X\in X^*$ with $\|y^*|_X\|_{X^*}\leq \|y^*\|_{Y^*}$. Suppose that $(y^*_n)_{n=1}^\infty$ is a sequence in $Y^*$ such that $\lim_n \|y^*_n\|_{Y^*}=\infty$ and $\|y^*_n|_X\|_{X^*}=1$ for all $n$. Can we find a sequence $(x_n)_{n=1}^\infty\subset B_X$ such that $\inf_n |y^*_n(x_n)|>0$ and $\lim_n \|x_n\|_Y=0$?

If the answer is no in general, is it true if $A\subset B$ are subsets of $c_{00}$ (the finitely non-zero scalars sequences), $$\|x\|_X=\sup\{|\langle f,x\rangle|: f\in B\}$$ and $$\|y\|_Y=\sup \{|\langle f,y\rangle|: f\in A\},$$ and $X,Y$ are the completions of $c_{00}$ with respect to these norms?