i have type 1 diabetes and have to deal w/ the low blood sugars, and i also have hypothyroid.
i dont think that you have to go back to eating animal flesh to be healthy/stable and even "traditional" RDs can agree on that (though, some, im sure, wouldnt).
the above suggestions, about eating often and whole foods is the way to go. consider adding healthy fats as well to your diet. high fiber and high fat foods and food combinations help stabilize your bs and sustain you longer.
ie: ever eaten "white" pancakes and been starving maybe 2 hours later? but if you had eaten a breakfast of pb on wheat toast, fruit, oatmeal....something w/ protein and high fiber, complex carbs... you are sustained much longer?
also, for example, low fat ice cream or fruit sorbet is hi in sugar, so a high fat ic would be a better choice for you.
breakfast is the easiest meals to make the most changes. you are usually eating in the same place each morning, instead of lunch or dinner where you may be out and about.... so make that meal ESP count.
again, snacking throughout the day is a good idea.