I don't suggest sleep aids. What time are you taking your Naturethroid? Are you splitting the dose?
It took me about 2 months to develop a pattern with Naturethroid - I had a few sleepness nights while getting adjusted.
Are you using any stimulants in the evening like coffee or soda? Most of the time when switching to a desiccated thyroid medication your adrenals need to sort out the new medication. Cortisol levels get off track. You could be higher in cortisol in the evening temporary - and things need to balance out.
A magnesium issue could be a problem too. Most thyroid patients do have some form of magnesium issues. A nightly supplement of magnesium may help things.
Switching to a natural desiccated thyroid medication from a synthetic - can be an adjustment. There are certain things that need to change to get the best positive effect out of it.
What are your current thyroid levels? FT3, FT4 and TSH? Inability to sleep is quite common if your levels aren't balanced.
You might also try taking magnesium and calcium before bed. They both have a tendency to be calming and magnesium can help with some of those "naggy" little aches and pains that we "hypos" deal with sometimes...
First though I'd recommend that you get your thyroid levels tested and make sure they are ok.
My suggestion is to ask your doctor about a non habbit forming sleep aide. Something like that might do the trick so you dont have to switch back. Also you might try meditation before you sleep to clear your mind along with having your significant other put on soothing lotions to calm your nerves. Just a suggestion. Maybe you had thought of it already but thats what I would have thought of if I was in your situation. Have a great day and I hope you get some good ZZZzzz's