I would use the plastic tubs.
Don't worry about having to wring them out. Have another tub and rinse, rinse, and then rinse some more.
Once the water is running clear you can use the washer/dryer without worry.
I worry more about getting some dye left over somewhere in the washer I can't reach to clean out and having it stain other things than staining the washer.
I did green and brown on a roll of burlap (which presents it's own washer issues, although I think I did end up making a mess in the washer once duck season was over), they came out good.
I dyed a dress shirt yellow and it really came out good.
I don't know what blue it was but when I tried deer hair/skin it came out very light.
I'd look for tips on getting dark colors to come out true.
Also if these donated sheets aren't the same fabrics you'll have issues matching. Try to separate them by fabric for your different colors.
Petroleum based knitter, removing that nasty oil from the ground one skein at a time.