Thank you for writing.
No chance of contracting any disease from having blood drawn. They are very sterile and know how to do it safely.
It's actually rather hard to get HIV. It takes unprotected sex to transmit it. (Even that is easy to thwart with a condom.)
And HIV is treatable. I've known someone who's had HIV for many years and is just fine.
Fears on the other hand, don't necessarily respond to reason and logic. Try yoga, qi-gong, tai-chi (meditative motion), or plain mindfulness meditation. 'Headspace' is one app that teaches mindfulness. There are some Headspace videos on YouTube that explain the concept. These exercises can help train the brain, strengthen various parts of the brain, shrink the Amygdala which causes fear, so anxiety lessens as a result. Takes a few weeks to start seeing the results. Worth a try.