Put blood in the fridge. The vet should've used RED TOP tubes, or red/black speckled top tubes (serum separator). Purple tube tops WILL NOT WORK. Your blood will be just fine over the weekend to ship on Monday.
WADDL can centrifuge to separate serum, don't worry about it.
I take a piece of papertowel and tear it into about 3 strips. (I use the little 3ml blood tubes for disease testing, the bigger 10 ml tubes will need more paper towel.)I wrap each tube in paper towel strip, and secure with tape that has a little tab for easy removal. I then put all the wrapped tubes in a small qt ziplock bag and put them in the fridge.
I keep little GEL freezer packs in the freezer. When I'm packing the box
right before taking it to the post office, I put an ice pack in a gallon size ziplock. I then snuggle the package of blood tubes right next to the ice pack in the gallon size ziplock, and fold it up. I then put the entire package into a styrofoam 'bait box' (for fishin' worms) which you can get cheap at your local bait shop. I then pack that styrofoam box into another cardboard box I'm sure to have laying around and pack it securely.
If you send package priority (cheaper, which is what I do), be sure to pack the blood in a styrofoam box with a gel ice pack. If you ship overnight, still use ice pack but you should be OK in a well-packed box.
Drawing blood is SUPER easy. I've been doing it for years - believe I was 15 or so the first time I drew blood - and nobody EVER taught me - I just read online how to do it. My dad restrains them and I draw. If your vet made it seem harder than it was, he's either ensuring you'll have him out next time (job security) or is inexperienced in drawing blood completely...