Bear in mind that it is possible to be overweight and, simultaneously, not getting essential nutrients that your body needs. That feels...well, pretty much how you describe you are feeling, ime!
I am pretty sensitive to not getting certain nutrients. They don't all show up on tests. Good if you can have iron, vitamin D, potassium/sodium (this is a test for other conditions) iodine, calcium, magnesium, and any others which affect mood.
For me, it's a combination of things that helps, and thyroxine is only a small part of it. (I'm now on a very low dose. It took many frustrating years to get a diagnosis and about a year to settle on the right dose. It is not sensible to take a different amount to that prescribed, but I am allowed to take an extra 25mcg every few days.) I need daily iron (my GP has cut my prescribed dose, but is ok with me taking more iron if I feel I need it and have physical signs.) I make sure we have red meat in our weekly diet, so the rest of the family are getting enough iron too. I also find iodine in my diet helps my mood enormously. Now, an iodine overdose is very nasty and dangerous, so I tend to just eat a bit of seaweed now and then rather than take tablets. There is a lot of iodine in some seaweed. I take a multivitimin a day if I notice any other problems.
However, I didn't do any of this (apart from eating red meat) when getting blood tests because I needed my basic body function to be assessed without help. Very frustrating. I truly feel your frustration. Do stay calm and be patient. You will get to the bottom of this and things will improve for you. Taking control of your diet is a great thing. It took me a year to lose my extra weight though. Even so, I ate very little each day, and I still put weight on very easily. Managing your diet for optimum fitness is something to do for life. It is a medical necessity and you have to deal with it yourself, a GP won't help much until it gets so excessive it threatens your life (morbid obesity).
Something else which has helped is finding out fairly recently that I have mild hyperextension of joints, which leads to sprains and pulled muscles, joint aches and generally makes exercise a lot harder, but much more necessary in order for me to stay healthy and fit. I now exercise daily as a form of self medication, to strengthen my legs, arms, joints and core (stretches and indoor exercise in winter, building up to short runs in better weather).
Losing weight and keeping the weight off is no easier now that I have daily levothyroxine. It is hard, it always will be, and I have to put effort into watching what I eat every day. I have to exercise just to be in reasonably good shape, I don't have time to do more to run off a packet of crisps or a biscuit. My joints can't take the strain of longer runs either. So I have learned to see snacks as my enemy (but have to keep this attitude to myself as one DC is underweight).