The problem with labels is that they're often inconsistent in how strict they are - GOTS is actually one of the better ones, but it mainly controls growing conditions, not necessarily how the wool gets processed later. If everything really matters to you (growing AND processing AND fair wages), you usually have to combine multiple certifications instead of relying on just one.
As for brands: I'd be careful with Katia and Drops - they're cheap, sure, but especially with cheaper yarns it's worth checking where they actually come from. More interesting are smaller labels like Naturally Yarns or regional suppliers that are transparent about where their wool comes from. It costs a bit more, but not so much more that it becomes unaffordable. Another approach: look into regional wool producers near you - some sell their wool directly or through smaller shops, and at least then you know exactly what you're getting.
Honestly, "really sustainable and cheap" is something that rarely works out - if the wool is very inexpensive, corners were probably cut somewhere, whether in wages or environmental impact. Your awareness of this issue is already a good start. Maybe don't switch everything all at once, but gradually try out different brands and see where you feel most comfortable.