If you already have an appointment for a physical scheduled, you might as well ask your doctor to order a blood panel for you. On the WesternU website, there should be a list of documents that is required. Did Western email you with what is required in terms of titers, etc? If so, print out all the required medical information pages and bring them to your doctor when you go in for your visit. Ask him/her to fill out a blood work order form. If you have insurance, there is usually no charge for blood work. I had my doctor hold onto all forms until my results came back, had him fill it out, give me a copy of the lab work, then scanned and emailed everything to student health. It really is just an extra two steps to get everything done. If you want to expedite things and your appointment is somewhat early (before the lunch hour), just don't eat anything before your appointment, let your dr. know that you have fasted and see if they can send you somewhere to get your lab work done right after your physical.
I know in past threads, students have mentioned that there is and isn't a ranking system. I don't think anyone knows for sure but considering that today is already June 1st (and one person in the FB group let us know that she has declined her spot in our cohort), it may work to your benefit to just send at least your health records in.
Good luck and I'm sorry that they're taking so long to start pulling people off the alternates list =/