Welcome to our message boards! I'm glad you found us.
The three month test for chlamydia and gonorrhea should be indicative of your result (negative). So, that's great.
You can take an HIV test today and be able to rely on that result. Most people who have it have a positive test by 3 months. If it's negative at 9 months, you can move on and not worry about having been infected from that encounter 9 months ago.
A lot of newer doctors have been trained to give an HIV test upon request (and without questions) or make it a part of the yearly exam. Even though it's not probable that you got it, it sounds like you wanted to know for your own piece of mind.
You mentioned some anxiety. The best way to calm it is to get tested so you'll know and won't spend another minute worrying about it. Your county health department can probably do a test for free or at low cost.
We're here if you need us.