I have HAND washed pieced strips after my cat anointed them. They did fray some, even with very careful handling. Because I wash my fabrics before cutting them, I was not worried about the sizes changing.
You can do all the things that you suggested.
You might want to wear protective gloves while you are washing the pieces.
What may happen:
Sometimes nothing except the fabric gets wet, the chemicals get washed out, and it dries.
Some of the fabrics may shrink or distort - you might have to compensate for the changed sizes - usually manageable if one is aware of the need to do so
There are people that think one can force fabric into submission by blocking it when it's damp. I think it's in the same category as expecting a rubber band to stay stretched after one let's it go back to its unstretched size.
I've washed little pieces in the sink - then put them in the washer on the spin cycle to get some of the excess water out - and then either hung them on a wooden drying rack or laid them on towels to dry.
It is a bit of a bother - but so is making a quilt or wall hanging or place mat when one can go to the store and buy one already made.
Guess it depends on how much the unwashed fabric is bothering you.
Maybe you can take some antihistamines or something until you are finished with it?