I'm so sorry. Those are so hard.
We have been going through this for about two years with my daughter. When it started she was almost four and now she's closer to six. The first few were really REALLY horrible.
Some things that help -
I don't lie to her about blood tests. She knows if we skip her morning meds, she's going to have blood work so there is no use lying to her and losing her trust.
We go to the same place every time and ask for the same lady. We found one who just strikes exactly the right note with my daughter. She stays calm, but she's not distant. She's not so sympathetic that she feeds into the frenzy. It's perfect. It means driving an hour each way but it's worth it.
We practice at home. At first, we couldn't even talk about it because she would freak out. Then she would "practice" on my husband and me. She asked for a doll that came with doctor stuff and she practices on the doll. She practices on the dog. She practices on her cousins. Each time she does us, my husband and I say things like, "Oh, I'm very nervous and scared but I've had this done before and I know it's just a pinch. I know if I'm REALLY STILL it will be over faster. I'm going to take deep breaths and make floppy arms" and we do the whole thing. We practice breathing with her and we practice making floppy arms.
We try to schedule something fun after the blood work so we can say, ok, let's get this done really fast and then we can go do Whatever It Is. The nurses will ask her what we're going to do and really talk it up.
She still cries. She gets REALLY upset. But she doesn't go totally rigid or fight them anymore. She doesn't throw up or hyperventilate so I feel like that's HUGE progress. The last one went really well. She still cried HARD the whole time but she didn't beg (oh, that kills me. That just breaks my heart) and she recovered quickly. In the past, she would manage to pop or pull the needle out every time they go it in so she'd come home with all these terrible bruises. She recovered faster, too.
I am sure it depends on the child, but talking and talking and trying to wait till she calms down does not work. It's easier to just get it over with and kinder, because she is not going to calm down, she's going to ramp up to a full blown actual panic and puke attack. On the days when she needs to be held down, I just get them to call in the extra people and we do it. I hold her hand and talk in her ear and try to remind her to breathe deep and relax, but there is just no point in trying to talk her around to being ok with it because we're just not there yet.
Good luck. It is so hard. I think a big part of it is talk about it a little at a time and take some of the SHOCK out of it, practice and role play. I guess it's a kind of desensitizing.