Having heard the horrors of military moves, I always elected the do-it-yourself moves when we were active duty. It can be hard work, especially when one is not able to work full steam ahead.
For many of the reasons stated, movers can't ship owner packed items. Here are ideas which may help. Rather than tubs, pack in plastic bags or plastic wrap. Zip Lock has those huge bags, which measure about 13X16. (available at WM.) They are excellent for packing individual projects. Burp the bags as much possible. Tie or tape the bags together in bundles of 5's and 10's. Get a large roll of cling wrap from Sams, Cosco, or a restaurant supply. Bundle your fabric groups just the way you want them, and wrap with the cling. Label as desired. Put everything back on the shelves or in drawers. Go back with a permanent marker and sequentially number everything. Keep the sizes of the bundles manageable, for your well being, and the convenience of the movers.
On packing day, the movers will grab packages of different sizes to fill the spaces in their boxes. Their inventory sheets will bare the numbers you assigned to the packages. On the other end, the bundles go in the drawers or shelves, but in no particular order. However, because each home has different lay-outs, you may want different arrangements anyway. Still, you can then easily put the things in the sequential order that they are now, and use only one arm.
Roll the tools, such as scissors, rotary cutters, craft knives, etc. in hand towels and wash cloths and tie tightly, then put into baggies. Tie books in small bundles also. For the sake of weight and space, the packers may put a tied bundle of books, and several bundles of fabric, and a measuring cup set from the kitchen. It may not look logical, but the boxes need to be filled tightly so that they'll support the weight of the boxes which get stacked on top of them. If one of your baggies pop because it wasn't burped adequately, things may begin collapsing.
If you have little utility drawers with snaps, machine needles, seam tape, and other what-nots. Stiff the draws full with sheets of bubble wrap or paper towels to keep the contents stationery. Wrap with the cling to keep the drawers closed. You could use tape, but then you'll need the goo-gone to remove the adhesive.
Take it easy on yourself, but give yourself peace of mind too. By wrapping in small bundles, you'll be doing a lot of lifting, but not heavy lifting.