@craigmorris, Greetings to you, and Congratulations on your liver transplant! I want to extend a virtual handshake to welcome you to Mayo Connect.
I also was prescribed Omeprezole after my liver and kidney transplant in 2009. I believe that it is a routine medication for us as we adjust to the new meds and to prevent acid reflux that can occur.
Craig, my first though upon reading your post is to make sure that you are in communication with your transplant team about your struggle with heartburn. Each of us reacts differently to meds and they might have some suggestions to help you. As a newly transplant patient, there is a tendency to 'not want to bother' our coordinator. Trust me, they are there to help us continue our recovery and new life. I transplanted at Mayo, and so I have a nurse coordinator and I use the patient portal to to communicate.
I don't remember many details, but, I learned something after my transplant that surprised me: no gall bladder. Yes, gall bladder was removed. At my 4 month check-up, I was instructed on healthy eating and the nutritionist stressed a heart healthy diet w/o fatty foods, fried foods etc. Of particular note is that my husband had his gall bladder removed several years ago. He had to avoid eating rich fatty food, and lying down too soon after eating.

I wonder if your heartburn is brought on by the food or food preparation. Do you eat many fast foods, or fried foods?