You need to snake the washer drain.
I have a bathroom and a laundry room that are next to each other. When the washer drains, the kitchen sink gurgles, then water backs up into the shower, then leaks from the toilet hole that I have poorly rigged with a cap.5 minutes later , water is gone from shower, but I have a floor to mop. You see this bathroom is unused. This bathroom also has a sink that is never used. The laundry room also has a sink that is never used. I have run a rented commercial snake from the toilet hole (it was the biggest and most conveinent hole) but it didn't seem to hit or unclog anything. The kitchen sink never backs up. The rest of the house does not back up. I ran a snake down the vents and they all went smooth. I understand I probably need to seal the cap I have on the toilet hole better. But this has this do it yourselfer climbing the walls.
The way you described it is your blockage is not on the secondary lines feeding the washer/kitchen, but further down the line. Snaking out the kitchen sink drains and through the washer standpipe will help, but you got something blocked further into the system or it would not be backing up into the shower.
What size machine where you using to snake the lines?
Will Rogers Plumbing