Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.
Start with a countable metric space X so that with one exception x, each point is open, but so that at the exceptional point, X is not locally compact at x.
It is easy to find such a subspace of the real line. ( start with 0 and (1/n)+(1/(m+n)) Now delete each 1/n).
Let Y be the one point compactification, adding to X a new point y, whose neighborhood complements are compact in X. In the new space Y, compact subsets are closed (and in particular Y is WH), but x and y are inseparable.
See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.