Suggest Remedy For Severe Cough

Hi I started noticing symptoms of me being sick around January 11. I had a sore throat and it hurt to swallow. Either the next day or the day after that I started coughing instead and my throat no longer hurt. Around the third day I was physically weak. I had been taking coughing nighttime coughing medicine to allow me to go to sleep (since I was coughing all night) and so we assumed that was why I was so physically tired. That night I took no coughing medicine and went to bed naturally. Again, the next day I was extremely physically tired. I woke up around noon (normal) and went shopping for about 2 hours. After first waking up I felt great but as those 2 hours went by I became more and more exhausted until I called it a day and we went home. We took my temperature but I never seemed to have a fever of any sort. Since then I have been pretty good during the day coughing wise (once every 5 or more minutes maybe) but at night and in the morning I am coughing a lot! Last night I even woke up in the middle of the night (rare) because of a coughing fit. I have noticed that sometimes when I cough little green pieces come up that I thought was mucus from my nose. I have looked online and I think that I have a infection in my lungs and basically I was writing here to get a second opinion and also see if you think I should see a doctor. I'm not huge on seeing doctors unless it is something that could be serious but in this case its been going on for a long time. If I didn't see a doctor and get medicine how long do you think this coughing could last? Thanks for your time.