25% to 50% of patients are hypothyroid a year after RAI. 5% more become hypothyroid each year and 90% are hypothyroid at the 10 year mark. The odds were very good you'd end up needing thyroid hormone replacement.
Some things you need to know. First is that TSH is a pituitary hormone, not a thyroid hormone. TSH is affected by so many things that at best it is only an indicator, to be considered along with more important indicators such as symptoms and also levels of the biologically active thyroid hormones, Free T4 and Free T3, which are not the same as Total T4 and Total T3. You only mentioned a couple of symptoms. Do you have others?
Before starting on that small dose of T4 med, I highly recommend doing some additional testing, to help further assess your status. Specifically I would ask to be tested for Free T4, Free T3, Vitamin D, and ferritin. With those results and reference ranges shown on the lab report, and additional info about symptoms we can advise further. For now, I see no pressing need to start on the med, but wouldn't suggest waiting for a few months. I would go ahead and try to get those tests done now. If you or the doctor have reservations about this, then I highly recommend reading at least the first two pages of the following link, and read more, if you want to get into the discussion and scientific evidence for all that is recommended. You an also share the paper with your doctor to try and get what is needed.
http://www.thyroiduk.org/tuk/TUK_PDFs/The%20Diagnosis%20and%20Treatment%20of%20Hypothyroidism%20%20August%202017%20%20Update.pdf