Good suggestions here - I also would have a problem getting rid of the fabric - I would rather get rid of the china and clothes!

when we moved here from california the few bolts we put in large trash bags and put them on the sides of the taller boxes as padding. my cup an saucers that are the only ones left from my stepmom coming by boat from france to canada in the late 40's after the war and thing s we didn't want to break we packed n the other fabric. it didn't get dirty so i just unwrapped it and re folded it. it had wrinkles but i always iron my fabric before starting to cut anyway.
that worked really well for us and not so much crumbled up paper to get rid of.

Does the new place have a dining room or L? You can eat in the kitchen!

I have moved many many many times ! Take some of the fabric and wrap it around the china. Wrap each piece of the china with the fabric. Take your fabric and put it in a trash bag and then put into boxes. Tape the boxes shut. This should do the trick of moving for you.