@parojulie, I have a liver and kidney transplant. My kidney failure was sudden and it as caused by my liver failure. This happened 11 years ago, so please excuse me if my memory of details is vague.
When I was in early stages of liver disease diagnosis, I took prednisone for a while, and then my GI added Azathioprine so he could reduce the prednisone. I don't recall the dosage or the length of time, however I do remember blood tests every 1-2 weeks, and at my 1st lab after beginning the Azathioprine, I bled heavily at the needle site and noticed that any bump or slight skin abrasion would bleed and bruise in a not-normal way. My GI had me stop the Azathioprine immediately. Currently Azathioprine remains on my list of 'do not administer' drugs.
I think I was mildly anemic with my liver failure, and for sure, with my kidney failure. While on dialysis, I received iron infusions and I remember a couple of times I was given a blood transfusion.
After my liver and kidney transplant, at my 4 month check-up, the kidney transplant doctor prescribed an iron supplements because I was anemic. I think I took the iron supplements for a couple of years. Today, I am advised to take a plain, no iron, daily vitamin.
I definitely urge you to keep in contact with your transplant team. You are still at the point where medicines will be adjusted as your body responds. Is your transplant team involved in your current care? Some of my favorite question(s) for my team: Why is this happening? What can be done about it? How worried should I be? And of course, the questions that you posted here are essential for you to ask them.