I have bronchitis and tried the many different at home remedies, but i still have a productive cough and a lot of phlegm in my throat. will i need to go back to the doctor and try to get an antibiotic? i have had the cough for almost 6 weeks now?

That's a long time you have had cough. It's time to see your doctor for a detailed history + chest exam so you can be treated accordingly. Please note that bronchitis is not the only cause of cough, post nasal drainage due to sinusitis can cause phlegm in your throat and cough. Some other conditions like reflux or asthma can manifest as such.

A sputum sample is recommended. TB testing as indicated. Asthma and GERD should be ruled out Consider a chest image. Your cough is chronic and the underlying cause should be determined. A Flu and COVID 19 vaccine recommended, unless otherwise contraindicated.