Hello Elwoodbear, after years of hypothyroid disease, I can tell you that double periods are a sure symptoms of low thyroid. I find it particularly interesting that the levothyroxin would prompt these double periods, as I've come to call them.
This is tells you that your thyroid disease is to the point that you likely have adrenal exhaustion, followed by low progesterone... All triggered by the hypothyroid disease.
Now, the fact that the symptoms heightened after starting levothyroxin is extremely concerning, as if the drug has pushed you over the edge. Please read the side effects of thus drug, as it is known to worsen thyroid disease. It actually triggers and worsens the autoimmune response responsible for thyroid disease.
Get yourself a NDT medication or ThyroGold supplements. Dump the kelp, as it is not the limiting factor and excessive iodine can be toxic. Worse, kelp supplements nowadays often contain high levels of radioactive iodine, which can worsen thyroid disease. So if you do feel you need a kelp supplement in the future, be sure to research your products.
The limiting factor in thyroid disease can often be amino acids, as thyroxin is composed of an amino acid bound to iodine. In addition, amino acids are required for the chemical reactions that allow thyroxin to work in the body. This is why in addition to amino acids, thyroid patients get relief from a high protein diet.
There are many diet and supplements that can help, as well as many that don't. Also note that thyroid disease can be caused by heavy metal toxicity from mercury and bromine. So cleansing and detoxing can be helpful.
My experience with the prescription meds was unsuccessful after five years and 8 different medications. I take essential amino acid supplements and find they are more effective than any medication.
I also care for my mother who has a history of thyroid disease, and she takes ThyroGold. It's unfortunate that she didn't have this simple OTC medication decades earlier.