I hear your frustration. I am concerned about this as well as in trying to find a job out of school after only having clinicals and no other experience, despite the program's reassurances to the contrary.
If there are no job postings, call and ask about per diem positions (to get your foot in the door), if this might be an option for you. Resubmit your resume including this in the cover letter for consideration.
I noticed that many are not using traditional postings anymore for phlebotomists and have to really reach out to find specific facilities to find postings. Not knowing what is in your area, in case you haven't already considered: hospitals, home health agencies, Red Cross, government research facilities, government patient healthcare facilities (VA centers), private medical research, doctor's care facilities/offices, outpatient labs, or nursing/retirement homes.
Hang in there. Sending good wishes your way. Good Luck!