As the daytime temps and soil warm up, your Peonies will take off. No need to move them. In fact, they don't really like to be disturbed. I've moved a couple over the years and it set back blooming for a year or two. They will basically thrive on neglect if they are happy in their location. I'm in zone 7a and most of the Peonies are just starting to set buds. Since you are seeing them popping up, they seem to be right on schedule for your zone.

You'll find some good information at the link below...

Here is a link that might be useful: The Old Farmer's Almanac

If you're in zone 5 and you're seeing cigar sized new growth I don't think yours are too deep! The Walmart bagged ones always take a few years, most are typically one eye if any. Hope you'll see blooms this spring! Some of my favorites were purchased from Walmart, it's worth the wait!