There is no health issue. Exposure to high levels of pure cobalt over a prolonged period of time is necessary for any problems to arise (e.g. breathing pure cobalt dust in a factory). Inhalation or ingestion of pure dust is most dangerous, skin contact causes the least problems. And this is NOT pure cobalt in a high concentration, high-risk environment, it's part of an alloy, which lessens any very small risk to practically negligible.
The same is true of many other elements that are used in common alloys in everyday life - nickel, magnesium, chromium, phosphorous, molybdenum, palladium, etc. - may cause problems in high concentrations over long periods, no consequence in small amounts. People don't drop dead from using stainless steel or phosphor bronze guitar strings. In fact I'd expect the risk with cobalts to be less than the risks with nickel strings (which some people have allergies to).
I'd be very surprised if the little shards you see are from the strings. My guess is steel wool. Many guitar techs use the finest grade of steel wool to clean fingerboards and frets. But it is notorious for spreading tiny thread-like shards that stick to pickups and strings. (It's the reason I don't use it very often - too damn messy!) If your tech doesn't clean his bench down very often, it's extremely likely that you'll get steel wool particles from a previous job on your guitar. If you ever use steel wool yourself, use masking tape to cover the pickups before you start and thoroughly clean down before re-stringing.
If you don't like Cobalt strings, that's cool. Strings are all about personal preference. Every brand and type of strings has it's own unique tone and feel. Find what works best for you. Cobalt is a new flavour in the marketplace, not everyone will like the tone or feel. The same is true of stainless steel, or pure nickel, or flatwounds, etc.
Next time please start a new thread rather than add to a different topic. It's just good "internet etiquette".
Thanks!