After reading all of that, my only comment is to say guess what? You're normal!
Every-freaking-body makes mistakes, and believe me, much worse than the mistake you made. As a new grad, you're focussing on so many things at once that you forget the basics (like taking vitals). As you get more experienced, you don't have to focus on tasks like priming iv tubing, because it's become second nature. That's when your brain will keep track of the things it needs to.
It's okay to feel terrible for this happening. That's your body's way of making you remember it so you don't ever do it again. I will never forget the day I gave 100 of lantus instead of 18. Thank God I caught it because the little old lady'd pry be in heaven right now if I didn't. Then again, maybe she'd be happier there...
One thing I'm realizing, (I'm a new grad as well, on Stepdown) is that being a new-grad nurse is probably harder than my entire nursing school experience. Again, this is normal. Normal, normal, normal. So stick it out! I PROMISE it gets better. You get better. You become the nurse you want to be. It just takes practice, which unfortunately, at times, involves errors.
If you feel like they're pushing you along too quickly, keep speaking up for yourself. And if they keep doing it, then go somewhere else! Every one of them has made errors before whether they told someone or not and they may have forgotten that. I'm on I believe my 10th week of orientation, and the end still isn't in sight, but that's because they understand the capacity of a new grad. (And it's because I'm on stepdown, but our med/surg nurses get 8 weeks and more if they request it).
All in all, don't feel like a bad nurse. You're not. You're just new. It really does take time to become a good nurse. You'll make it, just don't give up. Buck up!