Jayjay, lots of people feel worse when they start taking levo. I certainly did! It will probably pass as your dose increases.
I don't really think you're going to feel more and more hungry as your dose increases. It's possible that you weren't eating enough before you started your treatment. Hypos often have reduced appetites. But follow your body, it knows what it needs. This is not the time to be torturing it.
A TSH of 3.8 is too high. Hypos often have high blood pressure. Or did it just come up from low to normal? That is what is to be expected when you start treatment.
Yes, it will make a difference if you have it at 8 am or 10 am. TSH is highest early in the morning. You should have your test as early as possible. And you shouldn't take your dose of levo before the test, because that will give a false low reading. Take it after the test.
No, it is not alright to go around with a TSH of 11! You must take the hormone. You cannot live without thyroid hormones. Because the odds are - in fact, it's pretty certain - that it is only going to get worse. It is not going to 'sort itself out', as some doctors think. You need that hormone to live. Terrible things happen to bodies that don't have enough thyroid hormone! OK, think I've emphasised that enough! lol
Actually, you should insist on getting your FT4 and FT3 tested, as well as antibodies if they haven't already been done. They won't want to do them, but make a fuss until you do get them. Well, that's my advice. Others might say, save your breath and get them done privately. But that dépends on whether you can afford it.
It is, of course, possible that the levo doesn't suit you. It doesn't suit everybody. In which case, there are alternatives. Levo is T4, which has to be converted into T3. If your body has difficulty converting, for whatever reason, it's possible to take straight T3 - that's what I do - or a combo of the two. Or, there's NDT (Natural Dessicated Thyroid) which contains all five of the thyroid hormones. But you might have difficulty getting that from your doctor.
But don't try to run before you can walk. One step at a time - and baby steps at that. First find out how you get on with levo when your dose is increased. That could change everything.