No harm on pruning but I would do it next year, the more branches, the better for root development. Looks like they came from warmer conditions....which would explain the pushing of buds sooner.

My new bare root trees from Stark Bros in Missouri pushed out leaves way before my other fruit trees here in South GA. I thought it odd too.

Here is your local NC Extension information

Here is a link that might be useful: NC fruit growing information

I would certainly prune out narrow crotch branches and oversized ones, at least on the peaches. If a branch on the peach tree is more than half the diameter of the trunk you should definitely remove it as you can get an incredible amount of growth the first season if trees get ample water. Such a branch will suck energy from the rest of the tree and have to be removed next year anyway.

Cornell and Rutgers recommends pruning peaches shortly
after growth anyway as it reduces the risk of canker and doesn't slow them down appreciably. Unlike many other species, peach growth is not slowed much by pruning- partially because they grow vigorously throughout the season while apples and pears push most of their growth during spring surge.