Today was the big day of my semi-annual laundering of my wool tallit katan. As a preface, let me make it clear that I am writing this neither as a tzitzit professional per-se, nor as a launderer, but rather just as a fellow wool tallit katan wearer.
It's probably been several months since my tallit katan was last washed, but it had no smell at all - one of the advantages of wool. I removed the tzitzit and washed it by hand, the same way you wash any wool garment. I've often seen wool tallit katans that have a very different look because they were put in the washing machine. The fabric is altered and the fringes (assuming it has fringes) get balled up. I used to try to get my wife to hand wash my tzitzit, but eventually I realized she was really not up to it, so I learned to do it myself.
It's now hanging up outside to dry, and tomorrow (if I get around to it) I'll tie on new tzitzit strings, and it'll be good as new. If your tzitzit strings are not too frayed or dingy, it should be no problem just leaving them on when you handwash, but you need to make sure they don't get too tangled up, and I wouldn't leave is soaking for long.
I have two wool tallit katans, one for Shabbos and one for the weekdays. This one was the shabbier of the two, so it was for weekday use, but with the new tzitzit it will surpass my Shabbos one, which will be downgraded to weekday wear.
I wear wool in warm weather, but I also have some cotton tallit katans that I wear when it gets blazing hot. The laundering process is a bit different with those.