I have two girls, and yep sibling argue, I realised I can't sort it, they argue because they will, once you accept this and just say too them that YOUR not taking sides, it helps but that bit is the hard bit, to not get up and try to sort it.
Sit them down and tell them if they won't and do stuff for you, then your not playing games either, and stop doing things for them. The trip to the shop they want say "no, because you don't deserve the treat. "
It takes about a month for them to get the point, but it is worth it.
When I looked at where my month's money was going I realised I was spoiling the kids, because I am their mum and yet got nothing back, Eldest now has a paper round.
Your a great mum, your kids love you, you need to see that and stop being so hard on yourself.