Should I- 35F- see a pulmonologist, ENT, or allergist?

A little over a month ago I, 35F, got a bit of cold, main symptom was that I lost my voice for a few days and could not speak. Got over that, no problem.

About a week later, I got a wet cough (with mucus in the throat, nose)

I saw my pcp and she said everything seems good and that I must have had a virus that now has been "met with" allergies. Essentially it's a cough that's triggered by a feeling of mucus tickling the throat. When I do cough, I usually cough up a bunch of mucinex and it comes out my nose as well. Sometimes I get a huge coughing fit when I am trying to fall asleep. Sometimes I cough so hard that I gag.

I've tried Mucinex, Robitussin, Tessalon Perles, and Zyrtec. First three did nothing. Zyrtec probably helps the most with stopping the itch in my throat somewhat but not super effective.

I'll experiment with allergy meds if I know for sure that these are allergies but I am not sure. I never had allergies before and I don't have itchy eyes, a runny nose, or trouble breathing. I want a doctor to check me out and diagnose me.

I am 35F, 5'3, 125 pounds, do not smoke or drink. No medical issues.

TLDR. I've had a wet cough for a month. Is it best to see a pulmonologist, ENT, or allergist?