All right, so there is some more progress here! I added the ACSET Each target action instead of just the Action Effect, and my character now actually plays 2 animations now when she attacks! Hooray for progress! Haha! So now, the only thing that doesn't work is that she is still causing both instances of main-hand damage, since she's dealing 2 ~10s and not one 10 and one ~1. I have one animation set to a slash, and the other lightning, and both animations play the way you have in your gif. Adding the <dual wield skill> skill did nothing to the animations playing or altering the damage - both swings still do mainhand damage.
Interestingly enough, changing the ACSET to All instead of Each plays one animation and shows 2 damage numbers (both mainhand) like before, but Each does the two animations the way you have in your gif.