HIV DNA Testing

Good Evening all. Three weeks ago tomorrow I had unprotected sex with a high risk partner in a massage parlor. I was recently diagnosed with Chlamydia and  Gonorrhea. I have taken antibiotics for the Chlamydia and will be receiving a shot along with pills for the Gonorrhea on Thursday evening. I  extremely worried about testing positive for HIV as well. I have developed what maybe symptoms or maybe me stressing ie sores on my gums which are primarily on my top gum and not painful. In addition I have noticed small bumps on the tip of my penis only noticeable when erect. Since I have these symptoms and a high risk encounter I would like ringer HIV tested asap. I have read about DNA By PCR testing and have read that it can be accurate by 21 days. My questions are would you recommend this type of testing, would still having Gonorrhea affect the result and what is the likelihood of a false positive with this type of testing?

Thank you for any help or insight you can provide. I sincerely appreciate it.

[NoFiller]

Good Morning. To make things worse I had a very heavy occurring of night sweats last night. I was covered and my sheets and blanket were soaked through.