I'm sorry your concerns got overlooked till now. I hope you haven't given up on us.
Your experience with having to call for your blood results is why I never
accept any office policy that says they'll only call you if something is amiss. Results often never get sent by the lab, get lost by the MD's office... In one case, a friend's abnormal result got filed in someone else's folder. You are entitled to those results by law; they don't have the right not to inform you what they are.
You already have your diagnosis. That's a great start. And you already understand that there is no instant fix for this. You should
start to feel a change any day now, but everything depends on how long you've been hypoT and how your individual physiology responds to the replacement hormone.
Although most GYNs are capable of the diagnosis, most don't prefer to be the treatment provider. Maybe yours is... Maybe not. But then, most GPs are often not too great, either. So keep doing your homework. It's the best way to insure you get the right treatment to restore your health.
If the GYN refers you to an endocrinologist, please understand that they're generally far worse the the worst GP at treating thyroid. There are a few good apples in that barrel, and you may luck into one of them. Just be prepared to move on in case you don't.
You also already understand that you need free T4/T3 levels drawn. Those are most important, as you probably know. Try to find an MD who knows it, too. Other than those, you don't need any other tests at this time. You just need patience for the hormone to do its thing.
I wouldn't, however, wait a full 10 weeks to re-test. Eight weeks is plenty long, and probably 6 weeks is best. If your free T4 isn't at least dead center of its range at that time, a dose increase is justified. Don't allow your dose to be guided by TSH alone. But then, I suspect you know that, too.
You'll get your energy back in time. You just have to make sure to not let an MD tell you when you're "normal" again. Only you know that.
Best of luck! (And again, sorry for the delayed response.)