Trisk- I launder my shaks in the machine as well, on cold, gentle cycle, with some Woolite. Then I tumble dry on LOW for the smallest time possible, with a dryer sheet because that's the only way the cat hair comes off my clothes.

If you wash by hand, be sure to NEVER make scrubbing motions to wash it. Just swish it in the water like a tea bag and press gently to remopve water- no wringing. Put fresh water several times to rinse, and never scrub or wring the wool. Then press the excess water out gently (or put in the machine gentle spin cycle for a minute), and lay out to dry.

My advice?- use a lint brush in whatever direction works. I think you should expect all wool to develop some sort of minute fuzzy "patina" surface. It's wool, remember- woven from natural fur fibers. I wouldn't call it pilling though, it's just a softening/fuzzing of the surface. That's as much as I can guess.
I do notice little teeny fuzz balls occasionally accumulate along the flatlock seams of my shaks, but they just pull off easily and there are not that many of them at one time to be a problem.