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 (well, she was.. but that the Seanchan have no plan to kill her, I mean).

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.

In any case, bound at the block and the axe falling toward her head could symbolize a death threat approaching, which Egwene is incapable of stopping herself. She must hope for those who are running to her. Those who run may have discovered the woman Egwene is with is Mesaana. They know Egwene is alone with that woman and so is in mortal danger, especially that Egwene may trust that woman, and not realise her danger until it's too late. The fact she's bound to the block may symbolize the third Oath, that will prevent her to strike unless she's convinced this woman is BA or Forsaken. Those who run to her may, or not, need her warder to locate her. If Egwene is simply in her study with Mesaana, locating her may not even be an issue.

At this point this most likely is linked to a revenge by Mesaana, or by BA (Mesaana being a lot more likely). 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, with the variants that the Seanchan would capture her in the attack and kill her as a runaway damane.

But the main thing about Dreams is "what you see in the dream" is very rarely "what you get in reality."

One last point: there's is actually a connection between "justice" and Mesaana. During the WOS, she had "Mesaana's Children" capture people in charge, scholars etc., put them on trial and execute them. The dream may be a reference to Mesaana's idea of justice... making Egwene pay for ruining all her plans.