Let $X$ be the space $(\omega+1)\times(\omega_1+1)-\{(\omega,\omega_1)\}$, using the product topology. Thus, horizontally, it consists of a bunch of convergent sequences, converging to the right-most column; vertically, it consists of convergent transfinite $\omega_1$-sequences, converging to the top row; and we have specifically removed the upper right corner point. This is a Hausdorff space.
Let $A=\omega\times\omega_1\subset X$ be the points of these sequences, without their limits. Any countably infinite subset of $A$ is bounded below a countable ordinal and has accumulation points in $X$: either it contains infinitely many points from one coordinate, in which case it accumulates on the limit supremum in that coordinate, or it contains points from infinitely many coordinates, which will converge to a point of the form $(\omega,\alpha)$ at the right-most column in $X$. The closure of $A$ in $X$ is all of $X$, but this includes the top row of points of the form $(n,\omega_1)$, and this set has no accumulation point in $X$.