I’m so frustrated I’m honestly considering making up references and fake email addresses

My travel agencies have been so incompetent with getting and retaining the information for my references. They claim that have sent them emails and attempted to contact them multiple times. My references say they have gotten nothing. Then the agency asks me to ask my reference to email and text THEM to give me the reference. This is crazy. I am not going to beg my busy charge nurses to contact my agency on their off time. It should be as painless and easy as possible.

To make matters worse, when I finally did track all my charge nurses and managers all down and beg for them to directly contact my agencies (plural), the agencies rejected some of the references because my charge nurses forgot to check the box that said they directly supervised me. And the agency asked if I could ask them to contact the HR department again. When I applied to another PRN job 2 months later, they said I needed ALL NEW REFERENCES and needed to do the whole thing over again. It should not be this hard.