We're a family of 4 (2 adults, a teen son and daughter), and have never packed more than 1 22" carryon and 1 personal item (backpack or tote bag - bonus points if it doubles as a park bag or beach bag after arrival) apiece. We love the ease of traveling carryon-only, and once you get the hang of it, it's really easy.
If you're each entitled to bring a carryon size bag (22") for free, pack 1 carryon for each person.
If you're not entitled to bring free carryons, and it's the number of bags and not the size that is the issue, try to fit everything into two bags instead of three by packing lighter, using the tips below.
My suggestions:
1 - Pack clothing for half the trip length, plus one extra outfit (or at least an extra t-shirt) in case of a dining mishap -- that is, for 6 days. Add a laundry bag, lingerie bag, a baggie of wooden clothespins, and a baggie with detergent sheets and dryer sheets, and plan a relaxing pool/laundry afternoon midway through your trip. Every Disney resort has guest laundry facilities, and you'll usually find them conveniently located near the pool(s). If you're staying in a DVC villa, laundry is free: otherwise, bring a credit card.
2 - In each person's carryon (or in whatever luggage you decide upon), pack 6 warm (or slightly cool - check forecasts before you go) weather outfits, swimwear if needed, 1-2 nightgowns/PJs, socks and underwear, and extra footwear -- depending on the season, either 1 extra pair of cool-weather walking shoes that can be stuffed with smaller items like socks, or 1-2 pairs of sandals. Add each person's toiletries and meds in a small pouch, plus their 3-1-1 bag, keeping all contents to a minimum and decanting anything bulky into a smaller container. Each person should wear all of their bulkiest clothing items (pants, sneakers, a hoodie or jacket that matches the other outfits they're bringing, hat, other outerwear, etc.) on the plane. When selecting items to pack, try to pick things with a similar color scheme so you can mix and match them after laundering, and opt for breathable, lightweight "tech" fabrics wherever possible, as they tend to fold down to nothing and don't wrinkle easily. Do not pack extra clothes: that's the quickest way to derail a carryon-only objective.
3 - If your budget allows, purchase color-coded packing cubes (a different color for each family member) and pack each person's clothes in them. (If you really want to get wild, make your own DIY folding board with some cardboard and duct tape that will fold each item to the exact dimensions of your packing cubes. Not that I've done that, but... yes, I have totally done that!) Not only will this conserve space, keep bags organized, and allow clothes to be unpacked and placed in a drawer in moments, but when you go to do laundry, you'll be able to fold each person's clothes right back into their cube(s), so they can be put away effortlessly when you get back to the room. Also - and this is a matter of personal preference - I like to layer clothes in packing cubes as complete outfits (shirt-shorts-undies-shirt-short-undies) rather than as types (1 cube of shorts, 1 cube of shirts, 1 cube of underwear). That way, when I'm getting dressed each morning, I can simply take what's on top of my packing cube, instead of rooting around in multiple cubes to put an outfit together. YMMV.
4 - Don't pack or bring anything that the hotel already provides (e.g., shampoo, hair dryer, coffee), and share what you can between family members so you're not all packing multiples of the same thing (e.g., toothpaste, sunscreen, facial cleansing wipes).
5 - Use your "personal items" for the plane (backpack or tote: either use them as your park bag after you arrive, or nest your pack bag inside of them for travel) to hold: (1) things you'll need easy access to, during travel or in the parks (e.g., rain poncho, sunglasses, mini First-Aid kit with Tylenol, Band-Aids and blister covers); (2) electronics, earbuds, and chargers; and (3) any snacks or breakfast foods you're bringing with you. Since the food will be gone by the time you head home, the vacant space can be used for souvenirs on the return trip. My son typically doesn't even bother to carry a personal item or a park bag, so I bring a packable tote bag in my carryon (folds down to sandwich-bag size) that can be pressed into service, and travel with him on the homeward trip as his personal item, if we have a greater-than-usual amount of souvenirs.