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.