On a side note, could Shallan even wield a Shardblade? I seem to remember them being quite heavy - works well with Shardplate because of that. Can't find reference to weight right now though. Szeth's one is unusually small and he also had Stormlight to boost him. That being said, Shallan seemed to think she could use it, which is why she tried to summon it at one point - that she shouldn't bother trying to use the Soulcaster since she doesn't know how to use it but does start summoning the Shardblade instead, even though she promised herself that she wouldn't use it. So maybe she has used it before.
Actually, Shardblades aren't very heavy (was said a couple of times I believe), I think the main issue is that they are large, so it takes strength to have good control over them, but they aren't actually very heavy.
And I note that there were swords on the walls of the room, those would seem to be good candidates for spur of the moment murder weapons. And he wouldn't have expected her to do it, so she could have hit him before he had a chance to react.
She said she killed him, when talking to the symbolheads, and they seemed to accept it as a truth, for whatever that might be worth.