Yeah, I wouldn't overly worry about gono. It's possible, but not all that likely.
Since this feeling you get only happens when you do something unprotected, I'd suspect it's anxiety. To save yourself the anxiety, perhaps using condoms for every activity or limiting yourself to committed, monogamous relationships with partners who's status you know is a good idea. You can get flavored condoms. :)
Amox is not one of the few antibiotics that will cure gonorrhea. https://www2c.cdc.gov/podcasts/media/pdf/AudioRounds_Gonorrhea.pdf
It looks like 2 grams of azithromycin will cure it - that's taken at once. How much and how did you take it?
Stop trying to push or force anything. If you get a discharge, you'll know it, and won't have to force or squeeze it out. Let your body heal.
It's possible that you could be at risk for trich, but the risk isn't well known for oral. If you take that med, do NOT drink alcohol on it. It will make you sick. (It should say that on the package if you got it filled.)
*UPDATE*
So no discharge or burning during urination. I have been feeling like a soreness or like a build up in my shaft but I feel like thats because I have been trying to push and force urine as well as trying to squeeze any discharge out of my penis but so far nothing. I was wondering if this might be indicative of any STI. Thanks again.
The amox wouldn't cancel it out totally, but probably would offer some protection, and you have a low risk anyway. Receiving oral sex is a risk for hsv1 genitally (which is viral, so no protection from amox or any other antibiotic), gonorrhea and syphilis. Both oral gonorrhea and syphilis aren't that common, so while it's a risk, it's not something I'd overly worry about, especially being on antibiotics. The azithromycin won't do much for either gonorrhea or syphilis, but it might help your strep heal faster.
As for herpes, if you don't already have hsv1, I wouldn't worry about that unless you get symptoms.