I worked for a moving company, and it was the best job I ever had. Beats doing payroll any day. I would not recommend using fabric to pack dishes, etc. The rule to follow is - the heavier the item, the smaller the box. Always stand dishes on end, not flat, glasses on end, not flat. If you have a large bowl or vase, fill it with paper. The paper will absorb a shock. Lay a layer of crumbled up paper, then say the plates, a layer of paper, then the smaller plates, a layer of paper, then the cups, etc. Just number the boxes and keep an inventory if the moving company does not.
Use one different colored box, we used red, and that will hold all the small items - such as - if you remove the harp from a lamp, wrap it in paper and mark Lamp #1, L/R, and put it in the red box. Wrap the lamp in a blanket or heavy towels, and stand upright in a box. I kept lamps together. Then the shades, use a different box, put the little shades in, separate with paper or a towel, then a bigger shade, then a larger shade. Put paper on the top. Seal the box and mark the box, lamp shades, do not top, meaning that nothing should be put on the top of this box. You will have to mark the boxes on one side, not the top because once the boxes are stacked, you cannot see what is in it.
I would use small boxes for my fabric, like books are packed in. If you are using the plastic bins, tape them shut with packing tape. If you can, a better way is to fold blank newsprint till it is about 4" wide, laid inside each other and pulled out, and use the tape on top so no tape residue is on the plastic box. That way you can write on the tape. You can buy new or used boxes from moving companies or get free boxes that are advertised on craigslist, or some from the wine store. I do not envy someone doing it themselves. I would go in ahead of the men, strip the house so to speak, pack the dishes from the kitchen and dining room, then the lady's clothes, then whatever I could to finish my day out. Pictures are always packed with wrappings, placing them back to back, and front to front to avoid any puncture from the fixtures. If I can be of help to you, please send me a private message. Good luck. Our company used to unpack in a day, and I had all the dishes in the cupboards, the clothes where they belong, and all the furniture placed. All the client had to do was hang the pictures. I am serious. It can be done.