Your list of “important” papers might include: birth certificates, school records, credit receipts, social security card, recent bank records, current bills, phone lists, closing papers, realtor info, and more. Don’t leave these with the movers. Keep them with you!
Use brightly colored storage tote boxes, one for each person. Let each family member fill theirs with items they’ll want ‘right away’ in the new home — a set of sheets, a towel, a couple of extension cords, a phone, nightlights, address book, pens and paper, keys, Kleenex, and travel cosmetic case, and so on.