Could my thyroid cause big changes in blood glucose readings or would it be my new medication?

It is probably a combination of both.
Your body will need to take time to adjust to the new medicine.

Diabetes and thyroid disease are both endocrine, or hormone, problems. When thyroid disease occurs in someone with diabetes, it can make blood glucose control more difficult.

If you have hyperthyroidism your metabolism quickens and medicines go through your body quicker. Your blood glucose level may rise because your usual dosage does not stay in your body long enough to control it.
Hyperthyroidism and low blood glucose can be hard to tell apart. If you are sweating and having tremors from hyperthyroidism, you may think you have low blood glucose and eat extra food, causing your blood glucose to rise.

If you have hypothyroidism your metabolism slows, your blood glucose level may drop because your diabetes medicine doesn’t pass through your body as quickly as usual and so stays active longer. In hypothyroidism, it is often necessary to reduce your dose of diabetes medicines to prevent low blood glucose.
Hope this information is helpful.
You should see your doctor again if you still have problems controlling your blood glucose.