You give a classical story for candida (yeast) infection, particularly the combination of itching, superficial fissures, and scant "white pasty discharge" that adheres to the labia. I have to think candida is a strong possibility, even though your provider wasn't able to diagnose it. Hydrocortisone indeed would help the symptoms in the short run, but in the long run can help candida grow even more vigorously.
Herpes can mimic yeast, because it can cause soreness and itching. But herpes would come and go over several days, not cause persieting symptoms of the sort you describe. And there certainly is nothing in your description that suggests an active HPV infection.
You should return to your provider for reassessment, especially with candida in mind. Unless s/he sees something that suggests an STD, at this point I see no need for your partner to be evaluated. I doubt this has anything to do with him.
Best wishes-- HHH, MD
HHH, MD