First -- make sure no one will interrupt you and turn off the TV or radio -- it is so easy to get distracted.
When I take the washer and put it with the other pieces, I keep it in the same position that it was in on the cutter. I also put my pieces starting with the first piece off the farthest from me. I also have the new blade out and in the correct position for assembly. I put the blade I remove very far away from my work area (I hate to say how many times I put the old blade back in before I did this).
I have tried to use gloves when changing blades but I seem to grab multiple blades if I do.