I'm so so sorry you're suffering like this. It must be awful and I completely get how hopeless you must be feeling. But... I genuinely think that it isn't the levothyroxine that's making you feel like this. You aren't taking an amount that could cause the reaction you're describing. You get more exposure to chemicals from taking a dip in the local swimming pool or cleaning your toilet, even putting washing up liquid in a bowl of hot water. I know you're going to hate me for saying this, but I think the reaction you're having is panic-induced. Now, you might say, "how can this be panic, when it makes my heart rate rise so much?"
I had a long think about how to explain this, and eventually found an example of something in my own life. For years, I was terrified of having to give presentations. I'd force myself to do it, but hated every second - the build up to the event, the preparation, trying to think of all the awkward questions I might get asked, how embarrassed I'd be if I didn't know the answer. On the morning of a presentation, I'd have a massive tummy upset, sometimes just one end, sometimes being sick as well. And when it came to actually standing up to speak, my heart would be pounding, going horribly fast, my hands would be clammy, I'd feel slightly dizzy, sort of out of my body. Because I scripted everything, I'd manage to get most things out, but be horribly breathless, say words wrong etc. and afterwards I'd be utterly exhausted, wiped out for the next couple of days.
Something totally unexpected happened when I took a course of Prozac a few years back. I had to deliver a presentation and although I was still a little nervous, I found I couldn't get really worked up about it - that's how the drug worked for me, LOL. It kind of distanced me from, well, *me*. I couldn't be bothered to have all the thoughts about making a fool of myself. The presentation went brilliantly and I barely felt anxious at all. So did the next one (I have to do these things every few months in my current job). And the weirdest thing of all, even when I came off the Prozac, the next presentation was easy too. It was as though my brain had learned that I didn't need to be nervous. I've never found it much of a problem since.
I think - though of course I could be wrong (I don't know you) that something similar is happening to you every time you take a dose of Levo. You take it, and expect to have side effects - and sure enough, they turn up. It would be interesting to carry out a placebo test - to have someone switch your medication for something else without you knowing - to see what happened, whether you still had the weird reactions.
Please don't be offended by the suggestion that this might be all in your head. Because actually, most of our reactions to things are all in our heads. Hypnosis and neurolinguistic programming work on this basis. Paul McKenna wouldn't have sold any books if it didn't work. You can persuade hypnotised people to cluck like chickens, give up smoking, even lose weight.
I wish your doctor would take more pro-active action with you and not just leave you to suffer. It's not good enough to simply throw propranolol at you and send you on your way. He's failing in his duty of care to you. Hypothyroidism, if left untreated, can have horrendous effects as we all know - there's a reason people diagnosed with it are given free prescriptions!
I think it might be time for an open and honest chat with your GP to find out what he's really thinking and to request that he helps you to get well.