Yes, before the weather heats up and dries out (and when it cools again in fall) are prime shrub-moving times. Whenever you transplant anything, water it in well (as in DEEPLY) in its new spot, and keep an eye on it as it acclimates to make sure it has the least possible stress.
When there are rainy or cloudy days forecast just ahead I love to take advantage and transplant things, as they wilt less in those conditions.
So go for it…and know that they first few times you will find that you chop off a bit of root here and there–oops!–and so on, but most plants will recover just fine. Dig all the way around (do you have a fork and a spade?) before you try to pop it out…just feel your way, observing as you go.
You will get better at judging where to dig and how to lift things in time, with practice.
Slow-release fertilizers, such as an all-natural organic bagged product like the Espoma brand ones, are fine to incorporate according to package directions at planting time.