You are experiencing what is called "interrupted sleep" which is, most definitely, a symptom of hypothyroidism.
Since you are still experiencing at least one symptom of hypothyroidism, I suspect you are experiencing others. You might want to Google "hypothyroidism symptoms" and check ALL the lists - I think I counted over 60 symptoms total!
Bottom line, since you are experiencing hypo symptoms, this tells me that your thyroid care has not been optimized.
Many doctors are all-too-happy to dose a patient so their levels are in the normal range....but not necessarily the patient's normal. Most often, patients are undermedicated.
Now, I'm not sure what labs your doctor runs for you but, if it's only TSH, that isn't enough. TSH is a pituitary hormone - you really need to see what your thyroid hormone levels are.
Do you have a FreeT4 level you would like to share with us? More likely than not, if you are being medicated properly, your FreeT4 level would be towards the high end of the normal ranges - that's the "place" most people feel/function best.
Knowing your FreeT3 level would be helpful but, it's my understanding that the Royal College of Physicians sees no merit in this test....nor do they advocate T3 replacement (synthroid/levothyroxine is T4 replacement).
So, we'll work with what we have