Gawyn's PoV in The First Message clearly shows that Gawyn realises that there is only a rumour, and he only says that he would kill Rand if Morgase and Elayne were dead.

Later, in Like Lightning and Rain, he promises to allow Egwene to prove Rand innocent and to forestall judgment until then. Never mind that Egwene is an idiot and could have told him she was there when Rand learned of Morgase's death and decided to invade Andor to avenge her (News Comes to Cairhien).

And in Dumai's Wells he makes good that promise.

Not that I know which 'people who are directly involved in the incident and happen to be his close friends to boot' have told him otherwise.

But you know who does believe Morgase is dead based on a rumour? Everyone else. Rand invades Andor on the basis that he heard from Mat who heard from one of his men who heard from the enemy who have heard rumours that Morgase hasn't been seen in weeks. (News Comes to Cairhien).

Elayne also believes it because Nynaeve and Birgitte learned from Moghedien that Rand was attacking Rahvin to avenge Morgase's death based on the evidence I've described above. This evidence is apparently strong enough that the very idea that Whitecloaks and/or Darkfriends may be using her for political gain is considered impossible by her.