I used All Jersey Movers a few years ago. We packed the small stuff ourselves and had them package/wrap anything big/fragile but they will also do a full pack for you. They did a great job for our move.

Last time I checked the price difference between shoprite and case is only like 10 cents. I figured it was something like this though since they have all the same varieties in both brands, but didn’t want to risk it just to save a dime.