I think there is a strong link between adrenals and disturbed sleep, but I think it's actually the other way around... the adrenals are overproducing at night, and that makes it difficult to sleep.
Reading one of your previous posts, it sounds like this started after the RAI and before they put you on levo, so I think it would be premature to blame the levo for the disturbed sleep.
RAI is a pretty harsh treatment, and it may take some time to recover from it. It is done to deliberately damage the thyroid gland - but in order for the iodine to get to the thyroid gland, it has to start in your digestive system and work it's way through the body. I'm sure this must cause some inflammation, and your adrenals will be overproducing as your body responds to the inflammation and repairs the damage.
I think things should start to settle down as your recover from the treatment. I'm not sure how long this takes, but I have heard it can take a few months at least and that things go back to normal by about six months.
Also, since you've only just started on levo, you might not yet be on the right dose and that will make you feel pretty poor. One of the things you need to do now is make sure that your thyroid function is tested regularly and your dose adjusted until it's where it needs to be and thiings are stable. RAI is a bit hit and miss, they don't know how much thyroid function will be left after the treatment... so this is really important.
One final thought, I had a family friend who had RAI for thyroid cancer and she developed food intolerances as a result. Food intolerances involve an immune reaction that's going on in the digestive system and that can lead to strange adrenal goings on. My own problem is gluten and yeast intolerance and that has led to some pretty bad nights sleep.
Hope this helps and you feel better soon.