I took a quick listen to the song you mentioned. I am not a particular fan of Ed Sheeran, but the following might help:
I would suggest you listen to Planxty. I think they would be considered one of the best traditional Irish groups. I would suggest The Planxty Collection. You could also try some Christy Moore. Christy played in Planxty, and went solo when they slpit up (they split up a few times, but you get the idea). For me, you can't beat Live At The Point.
I would also suggest you try stuff by The Dubliners. I would suggest The Very Best of The Dubliners.
From there, you could listen to The Pogues. Rum Sodomy and the Lash is good, as is Hells Ditch. But you won't go wrong by listening to a Best of, or Collections type of album.
I would also suggest listening to the Dropkick Murphys (Do or Die or Sing Loud, Sing Proud!, or Flogging Molly (Drunken Lullabies). These two groups and mostly American, but with a very Irish influence. They have a more punk element in them too.
Hope this helps,