I don't make lace often. But here is my take on this. I always wash items before steaming. Cause I don't want to risk setting in any dirt, hand oils, etc.
The key when washing wool is technique. Don't wring it, agitate it, use overly hot water, or harsh soaps.
Here is my technique. Fill the sink with lukewarm water. Add a squirt of shampoo or delicate soap. Sloosh to combine soap. Lay in the wool scarf. Leave it sit & soak for 30mins. I takes that long for wool to get fully saturated. Just let it sit there-don't squeeze, wring etc. It comes clean on it's own. Drain water, press out excess & refill with clean lukewarm water to rinse. Let that sit for 5mins (you need less time cause wool is already wet.) Drain & repeat til water runs clear. Now gently press out extra water with your hands. Roll in a towel to get it damp dry.
Move to a blocking board. Stretch & pin when damp. Your pins are rust proof right? They should be, but just checking. Let it air dry. Then go back & steam. The steam is to block the stitches & set them in place. The iron doesn't touch the work. Just the steam.