We've only left our children like that twice, and they are now 5 and 7. My husband travels regularly for business (the kids are used to him being gone), and we finally decided to take advantage of it and combine a refreshing vacation for the two of us at the same time. And I had the same separation anxiety going on (just me, my husband was fine with it). So I'll tell you how it happened for us, and hopefully it will give you some encouragement.
On our first trip, I was gone about a week. Our kids were 3 and 4, and I had to try really hard not to show them how anxious I was to leave them. And after I got on the plane, I had this super strange feeling like part of me was missing since we spend all day, every day together, and then they weren't there. But after a day or so I was feeling better and really enjoying myself, and it did wonders to have some alone time with my husband. And it turned out that our kids were having a blast with Grandma and Grandpa, and didn't have any difficulty at all... Until the day I got home again, when all of a sudden they remembered that they missed me and were all over me (in a good way), and had fun telling me of all their adventures.
Our second trip was a week and a half, and our kids were 4 and 6. They were with Grandma and Grandpa again, but they happened to be in the middle of a move, so there were fewer fun activities for them to do. But I was more confident to leave and let them have their "adventure". Our 4-yr-old did absolutely fine, but our 6-yr-old this time cried a lot and made a lot of frustration for the grandparents because of missing me. I think it would have been better if there had been more structured, fun activities to help him keep busy, rather than basically being on his own while the grandparents were busy with moving households.
So my advice for you would be to make yourself go and enjoy the trip. Talk to your youngest about how he's going to have an "adventure" with the grandparents and you'll be so excited to hear about it when you come back. Don't show your anxiety, or they will pick up on it and become anxious too. And make sure the grandparents understand that they will need some structure and planned activities and outings to keep the children occupied (not just passing the time in front of the tv, for example). I don't mean going to the amusement park everyday, but things like the park, library, nature walks, storytimes at the bookstore, playing with cousins.... I'm certain it will all turn out worlds better than you fear. Best wishes!