IRRC, the headsman isn't one of the ways the Seanchan put people to death. They would make Egwene damane anyway, not kill her. They already have an Amyrlin Seat prisoner, and Brandon all but confirmed she's been leashed.

There likely won't be an headsman involved for real either, there's always something important in the dreams that is wholly symbolic (the cliff in the dreams about Perrin, the lamp and ravens in the one about the Seanchan attack, the blindness of Rand in the wolfdream etc.). The imagery suggests capital punishment, but in a more general way it can simply be mortal danger, the fact the blade is approaching in slow-motion could mean she won't be facing instant death, but a growing/approaching threat. Egwene is also restrained/captive. She cannot stop the axe herself. She could have been captured, shielded and bound (in Air?) then.

At this point this most likely would apply to a revenge by Mesaana, or by BA. If either get their hands on Egwene before or during TG, it is possible they mean to put her to death slowly and cruelly (Mesaana has a mean streak, and she's not forgiving). Egwene's sent hundreds of BA to the headsman... We've yet to see in how much trouble the "Verin/Egwene fiasco" has landed Alviarin and/or Mesaana with SH/Moridin. Shaidar Haran certainly won't be pleased, he had punished Mesaana before for wasting the BA, and ordered Alviarin to find those hunting the BA. Mesaana may not even have chosen to remain around. She may have been forced to by Shaidar Haran, when (if) Alviarin reported to him abou the purge and was ordered to warn the BA and flee. Mesaana herself may have been ordered to stay and find an opportunity to kill Egwene at the right time, when TG starts, for instance. This would disorganize the Tower at a very bad moment.

In any case, Gawyn (possibly it's Gawyn) running to her suggests to me that when Egwene will be in danger for her life, he'll be able to sense it, much like Birgitte and Elayne. This speaks of them being bonded by that time (which could mean it won't be early in TOM, might even happen during TG). It may not be Gawyn who's able to deal with the threat Egwene is under (especially if it's Mesaana). He may be leading to her those who can save her. Notice Egwene seems to think so. She calls her potential saviour "they", not "he".

But that possibility aside, I really don't think there's enough story left for Egwene to be captured by the Seanchan (and certainly not for some big spectacular rescue....) - nor certainly for any story line in which Egwene would be held accountable for crimes under Tower Law.

Notice that Brandon said this Dream could still happen. Interesting, in the context that Dreams are just probable and not certain like Foretelling and Viewings. There must be one or two in the lot that won't happen. I don't think it will be this one that fails, but we never know.

I don't think this dream is connected to the one with "Gawyn's choice". IMO Gawyn's choice was fulfilled long ago in Cairhien. The choice was between his duty and Egwene. He chose Egwene. Had he chosen his duty, she fell in Galina's hands...

As for the headsman... I'm pretty convinced Jordan chose that symbolism because of its ambiguity, because beside what it will represent for real, it could fool those who intepret the Dreams too literally into believing Egwene would face justice once captured by Elaida. It worked. The theories were all over the place about how Elaida would send Egwene to headsman in TGS.

But the thing about Dreams is "what you see" is very rarely "what you get."