In TBC, it was made possible to change profession specializations without unlearning the entire profession. This was accomplished by going to the original trainer, paying gold to unlearn the specialization, and going to the book “Soothsaying for Dummies” in Tanaris.
Currently the book “Soothsaying for Dummies” offers no options after unlearning a specialization (in my case, Armorsmithing). In addition, I am unable to reacquire Armorsmithing after dropping it. The Weaponsmithing trainers also do not offer me anything.
Edit: See my reply below. “Soothsaying for Dummies” is not used to switch from Armorsmithing to Weaponsmithing.