Needle phobia as well and I'm a healthcare provider. Usually give them 2-tries. Currently, I have a picc line in my right arm with 2-available ports, no needles for those 1-2 blood draws, currently outpatient therapy for AML diagnosis in June.
You have this, and ask for someone else after 2-tries.

I feel bad when I see so many horror stories about blood draws, etc.

I have been getting two blood draws a week for nearly a year now. Recently, I have been getting platelet infusion, almost every week. I have never had a bad experience. In fact, I barely notice the needle stick.

I get all my care at Mayo in Rochester, MN. The secret is that the phlebotomists just do phlebotomy activities and, thus, have perfected their practice. On a 1-10 scale, I give them a 15+.

I had the same as you Phillip, had to have an ultrasound to find the veins.
When that started I said you have to find an alternative way to extract blood. I had a port put in my chest and now it’s a quick procedure . Ask to have that done.

I had them get a sonogram to find a vein my suggestion is to hydrate when you know your going to do a draw

Phillip There are a few things I do is to work on hydration. Ask for hot packs or heated blanket to warm up your arms. Also, you are the patient, you have a right to say no & come back another day. If you are havaing frequent draws speak to oyour doctor, maybe you need a access for lab draws & medications. Best wishes.