Bibiboo, you poor thing
IMHO, nurses just aren't too good at taking blood, as they don't do it too often. The patient gets anxious, they get anxious - bit of a vicious circle.
I presume that you are always lying down to have blood drawn.
Ask to be referred to your local hospital to get a plebotomist to do it. A phlebotomist's job is to take blood, all day. They never miss a vein and are absolutely superb
Alternatively, if you have a friend who is a hospital doctor, they can do it. You just have to get the midwife to sign the forms and give you the tubes and needles. This is what I did, after 4 failed attempts in two arms - and I really don't mind have me veins poked about in!
The last alternative is to get referred to the ante-natal clinic for a doctor to draw blood after you've had some local anaesthetic cream dolloped on your arm. They do this for kids. However, it is a bit of a faff as the cream takes a while to numb the skin.
Remember, for your 20wk bloods, think ahead and maybe request that the 20wk scan and bloods are both done at hospital on the same day - less stress for you maybe?
Finally, don't let anyone let you feel that you are neurotic. Many people are what medics call 'vasovagal'. Basically, pain makes you faint. My DH, who is a doctor, suffers really badly from this and always has to be horizontal if a dentist or doctor approaches him with a needle!
Good luck. I hope you can get it sorted soon and get down to the more serious stuff like shopping for maternity clothes