I highly doubt that Solimo is CF. It just does not align with Amazon's stance at all; Amazon is all about doing things as cheap as possible with as much profit as possible. Plus... they don't even care about the safety/wellbeing of their employees, so I highly doubt they care about animals. Until a reliable 3rd party confirms it, I wouldn't trust what they claim.
Vegan and CF coal tar shampoo doesn't seem to exist. I've searched high and low. :( The closest thing I could find is Heritage Store - Pine Tar Shampoo, which focuses more on healthy scalps than dandruff removal.
There are other options for dandruff shampoo such as Jāsön - Dandruff Relief, which contains salicylic acid and sulfur. Or options from Derma E, Avalon, Paul Mitchell Tea Tree (cult classic).
It's just one of the few products where we have very limited choices as vegan. It's really frustrating. Dandruff can actually be caused by a fungus on your scalp (most common) or dryness. If it's fungus then most of these milder options won't work. In the case of it being chronic or fungus related I would treat it like I would any medical issue and use the products I need even if that means they may not be cruelty free or vegan.
My bf uses this stuff but he does not have locs. Sometimes I put a bit of coconut oil on my scalp too.
OK sounds like seborrheic dermatitis. Diet isn't going to cure it. Medication will. The good news is there a long list to choose from. If one doesn't work then another one might.
All of these things can dry your hair out a lot. So you may want to try a leave in conditioner or a detangler. Kids' detangler works for me.