To do some digging, don't focus on dog diseases but research human thyroid problems. Dogs will be very similar. You'll get more information.
Thyroid can be a little bit tricky, because you can have too much thyroid hormone or too little. Each of those options has multiple possible causes. With each of them the thyroid can be too big or too small. The bulging eyes (exophthalmia) is due to deposits behind the eyeball which in most cases will go away at least partly, when hormonal balance is reestablished. My mother had an overactive thyroid problem (resolved with surgery) and in her case, one eye only partly resolved while the other one went back completely. Only she ever really worried about it, though.
Thyroid problems can be due to lack of iodine in the diet, or lack of the body recognising or able to use the iodine (which will cause en enlarged thyroid even with underactivity) or it can be due to the body's thermostat being set too high (at the pituitary and hypothalamus level) which triggers the thyroid into overproduction, and the thyroid enlarging just to cope.
As I said, there can be many different causes and you need a vet doing examinations, history and regular blood tests to find out exactly what is going on.
Once you have worked out exactly what the problem is, treatment is generally a matter of medication and monitoring. Exercising off the weight is not the way to go, as the vet has probably already explained. Getting the T3 & T4 levels right will make the biggest difference. I hope they're also testing TSH (the pituitary hormone) because in some cases that can be set wrong. The body is an amazing thing, though - the TSH levels should be automatically programmed by feedback from the thyroid, of the T3 & T4 levels. Imbalances occur when this feedback system is malfunctioning or the raw materials for the hormones are not available (as in iodine deficiency). A low thyroid will be making the poor thing feel sluggish. An overactive thyroid will make it nervous, jittery and maybe some loss of condition of coat and skin.
With both types of problem - if the thyroid is enlarged it could be causing discomfort with swallowing, which could affect how they eat & drink. I'm not sure if it affects the sound of the bark - it's a possibility. It should return to normal with treatment.
Thyroid is not such bad news - in comparison, adrenal hormone problems can be nasty.
Good luck with getting info. Also ask the vet for some information on it, he should have access to websites or leaflets to help you understand what is happening. You are about to get to know your vet very well, with the ongoing supervision your dog will need, although this sort of intervention shouldn't be too costly in canine terms.
Marg