You've already asked the HIV question in the HIV forum, but I'm sure they'll tell you there is no risk for HIV, and there isn't.
No STD gives you a cough, but covid certainly can. Have you been tested for covid?
If you have covid, or another illness, it doesn't increase the risk of getting an STD, but there is a risk for oral gonorrhea, chlamydia and syphilis when giving unprotected oral sex.
You've already had gonorrhea once. No judgements, but you should really consider using condoms. It's okay to insist on them. :)