You are relatively early in this whole process, so it should get at least somewhat better.
Of course, part of it will depend on how the D process goes, how good you are at just letting things go, etc.
Things to keep in mind - it may take a while, but kids do see through this kind of stuff, and generally do see who is really there for them. He is still their dad, so they are going to try as hard as they can to let him be part of their lives any way they can, but they do know the difference.
Also, remember that doing this isn't about whether or not he really is a good dad, or if he gives monetary support, or any of that. It's about letting your kids have an opportunity to have the best relationship possible with their other parent. If he falls down on that, it's on him.
All that being said, I do have a suggestion...
What is something you enjoy doing... something you have trouble finding time for because you are too busy being both parents to your kids? Do you really like to read? Any crafts - sewing, cross stitch, crochet, scrapbooking, or anything like that? Or something like that you want to try and learn but never seem to have the time to do?
Or maybe you never have time to really pamper yourself - a long bubble bath, do your nails, spa-type stuff?
Whatever it is, figure out something like that - something you can do for just you. Something you can really throw yourself into. And use the Skype time as your chance to do just that. Your kids are old enough that you should be able to make a deal with them - twice a week, you are going to set aside an hour for yourself to do X. It will be during the same time they Skype with their dad. If the Skype doesn't take up that whole amount of time, they are expected to do homework, read, play video games, or whatever else they should be/want to be doing. They shouldn't interrupt you (outside of an emergency) for that period of time.
Then do it. Get really involved in whatever it is you want to do. Preferably with headphones on playing some great music. Get so involved that you forget about everything around you for that period of time.
Focusing on doing something for yourself is a great way to get other things off your mind. You won't be hiding - you'll be busy doing something you enjoy and getting some (probably VERY needed) "you time".