If I increase the dose of Glucophage, is there a chance that I will lose weight?

No, simply increasing the dose of Glucophage will not significantly reduce body weight. Of course, the doctor is right that the correct treatment of hypothyroidism will contribute to the fact that it will be easier to reduce excess body weight. Please remember, however, that a caloric deficit is necessary for weight loss. Without it, even with proper treatment, the remaining pathologies - body weight will not begin to fall. Weight loss itself is the other side in this vicious circle - as a result of weight loss (if you are overweight or obese) insulin sensitivity will also improve and probably parameters related to thyroid inflammation (although of course Hashimoto's cannot be cured at the moment, it is possible to control the disease so that it does not hinder functioning). However, both ailments reported by you are very strongly related to the need to change your lifestyle. In such a situation, it is worth seeking help in the subject of depression, because it unfortunately makes it difficult to make lifestyle changes that are necessary here. I leave links that may interest you: https://www.gdziepolek.pl/artykuly/insulinoopornosc-i-stan-przedcukrzycowy-o-krok-od-cukrzycy