Keeping Busy...

You didnt say how many times you've watched these newbies, but I think after they get "used" to the way you do things they will adjust. You may need to nurture them "holding hands and whatever" for a little while in order to build their trust so they will feel comfortable. Sounds like right now they are just "scared" of the new environment, missing their own mom and such.
If your kids are stable, you trained them to be that way. With a little time you will be able to do it to the kids you babysit as well. It wont happen over night tho. Give them the extra attention needed and when they feel "secure" you will find them not being so needy.

I have a 5 year old and almost 3 yr old. I work a lot from home, as well as the normal routines of the day. My children had adapted, unfortunately, they watch TV a little much to occupy their time but they are quiet. I now am watching a 5 month old one day a week, and I am trapped inside my house to an infant routine. After 1 month of him being in my house, he has learned to adapt to my schedule too. It takes time.

I would keep doing what you are doing, your kids adjusted to how you do things they will too i time. if you give into those needy needs and your kids see it, your kids may go backwards, you don't want that. J.

Give it time. I watched a 6 mos. old that had to be held all of the time b/c mother/grandmother did that. After a week, the baby did not need to be held all of the time by me, she was very content in her pack-n-play when I got her bottles ready or had to put a load in the washer. Although, every Monday it took some readjusting but she did it.