If doctor shopping is what you have to do to find a doctor who is willing to listen to you, believe you, and work with you on a treatment regimen....then, by all means, do doctor shop!
Your main concern is getting appropriate treatment so that you can manage your symptoms and maintain some form of a relatively normal life. Who cares what anyone else thinks about you changing doctors. They are not living with your disease, so their opinion carries no weight whatsoever.
If your doctor is so unaware as to not know that it is the medications that are bringing your blood tests into the normal range, then he/she is not the doctor that you need. At this point, a knowledgeable doctor should be talking about adjusting you meds to keep your test results normal and to continue to alleviate your symptoms. So, in short, yes continue to doctor shop until you find a doctor who works with you, not against you!
I wish you the very best
Peace and Blessings