B9exchange
I regularly wash the linen covers on our sofas, they come out really clean, and I then put them back damp, straight out of the washing machine, and let them dry in situ, no ironing required.
Yes this was what I used to do on our old loose-covered sofa, though you do have to limit the spin speed to avoid creases. I used to air dry it for a while as it could be a bit to wet when it came out of the machine. The fabric will probably tighten up when it is wet, you'll need some help with the stretching over the chair.
I can't understand why the colour came out of the fabric with bicarb. It might be easiest to try to fade the rest of the fabric a little, perhaps a cold wash with a normal powder detergent and a little of one of those powder stain removers that you add to a wash.