... And paying his visit to the BT at the same time, noticing the dreamspike.

The Shadow really didn't want Rand in Caemlyn. Logain's flight to Rand was a close call, and the Shadow probably breathe more easily when Verin or Elza reported Rand rejected Logain's accusations.

Verin even interjected, and tried to deflect attention on something else: whatever he is up to, Master Taim has to wait because TG is upon us, yada, yada. Elza was present, I think.

It sounds to me that Verin got orders to report anything having to do with Rand and Andor, and to try to prevent any visit from him there. That may even be when she learned about the invasion, from the Forsaken who gave her these orders. Mesaana spoke of Elza has "our woman near Rand", and it may be a clue Demandred is involved with the plans for Caemlyn. It's big - we also know Moridin and Moghedien are involved, and the dreamspike is used to protect the BT etc. It sounds like a central element of the Shadow's global TG plans.

I think Demandred has promised to Moridin he can retake for the Shadow all of Rand's conquests in the opening move of the war. It makes the invasion of the Borderlands look like a bait/distraction to make the Light look to the north, not the South... It sounds likely the Light will soon be caught between two territories of the Shadow. This would also have isolated the Seanchan from TG, before the stole Travelling.