Does a dry cough for nearly three months indicate an HIV infection?

Hello doctor,

I had sexual intercourse with a sex worker 23 weeks ago. And it was protected and was my first time, and it did not last more than 10 minutes. After a few days, I fell sick for a day or two, and it was gone. But then my friend told me condoms are not 100 ℅ protective. So that scared me. For a few weeks, I did not have the guts to get myself tested, but in the 10th week, I got myself tested with an ELISA kit, which was negative. Then I read on the internet that card tests are not perfect, so on the 12th week, I took HIV 1 and 2 serum test, which was negative. After that, I took multiple tests, and all were negative. But what scares me is that I had a dry cough for nearly three months and consulted a doctor. He told me it was an allergy, and my X-ray report was normal, so he sent my blood for an immunoglobulin test to see what it could be. Was this cough due to HIV? I have been a smoker for five years. Can the IgG test show the HIV in it?

Thank you.