You did get a prescription, though? Right? He's not putting it off for a month? If he is, he's a quack and you need to see a different doctor and get medication sooner than that.
Rest may help some, but you're not going to feel better until your thyroid is back on track. It could take several months to get your dosage right. Cardinal rules: Don't alter the dosage yourself without your doctor's approval; Until you get your TSH in the normal range AND feel like yourself again, keep getting blood tests every two or three months so your dosage can be adjusted; Follow the medication instructions about taking it at the same time every day, on an empty stomach, not with certain minerals (like iron), etc.
I've been where you are. You can sleep 18 hours a day and it can still not feel like enough. But you do need more rest and to take it easy. If you feel you can take time off, do so. If you worry that it would damage your job security or something and you feel you can get through it, then don't. It's your life. You're an adult and you know how you are feeling better than anyone. If you feel bad enough to take time off, do it. Don't push yourself to the point of exhaustion.
I hope you feel better soon. I'm also very glad that they found out what was wrong and it's a pretty easy fix.
