Indecisive Cough question

Hi! Probably a really silly post, but...

I thought I'd just gotten over a heavy cold (I think I probably had a bit of a chest infection as well since I had a sore chest and was coughing up tons of green stuff (too much info XD)).

But now, although my cold seems to have gone, my cough keeps alternating between productive, dry and tickly during the day.

Just wondering if it's possible to have all cough types at the same time?

My throat & chest still hurt a fair bit and it's been 2 weeks (I've been dying to go to karaoke with my friends the whole time hehe).

The dry cough especially makes my chest really hurt.

My new company has been slow about getting my health insurance sorted, so I can't even go to the doctors if I wanted to at the moment.

So I'm sticking it out more or less alone...just me and the Japanese pharmacies!