It’s important to clarify that in a clinical setting, dandruff is associated with seborrheic dermatitis, which is not (and is not treated) the same as psoriasis. From UpToDate:
Seborrheic dermatitis in adolescents and adults. Denis Sasseville MD, 2018.
Scalp — The mildest and most common form of scalp ...
I can offer only partial answer to your question, and that is on how this shampoo works.
From The Electronic Medicines Compendium:
Mechanism of action
Coal tars suppress DNA synthesis in hyperplastic skin inhibiting mitotic activity protein synthesis and cell reproduction.
Coal tar decreases epidermal proliferation and dermal infiltration and thus ...
Malassezia overgrowth is thought be the cause of most cases of dandruff (1) and rotating treatments avoids the development of resistance. However, some have argued that overgrowth of Malassezia is a result rather than a cause of dandruff (2).
1 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4852869
2 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2887514
Currently, there seems to be no convincing evidence to say that alternating between shampoos prolongs their effectiveness in treating dandruff.
1) According to the Scientific Committee on Cosmetic Products and Non-food Products Intended for Consumers by European Commission, 2012:
There is at present no scientific evidence of development of