I hate to say it, but your diet is probably exacerbating your issues, especially if it's heavy in the processed vegan stuff.

First, make sure you're getting enough nutrients. Iodine, B12, and Magnesium are particularly important for a lot of bodily functions relating to metabolism and thyroid, and Omega-3 and other fats are important for overall body functioning. Many vegans, especially those that aren't aware of the potential nutrient deficiencies of the vegan diet, become severely deficient in these and other key nutrients.

Second, try cutting out grains and legumes (or at least gluten-containing grains and soy) for at least a month. Soy and gluten, in particular, are, at best, a mixed bag when it comes to their supposed benefits. However, the downsides can really screw with those of us that are susceptible to the not-so-great effects. Soy, especially, has high levels of phytoestrogens, which further disrupts our already-vulnerable endocrine system (many women with PCOS has chronically elevated levels of estrogen).

Furthermore, many women with PCOS have reported improvements to pretty much all PCOS symptoms after eliminating gluten, and the same goes for Hashimoto's sufferers. I've personally found drastic improvement in inflammation-related issues, including chronic headaches and migraines, and acne, by eliminating gluten from my diet. Not only are many PCOS and Hashi's people sensitive to gluten, but grains and legumes also contain a number of nutrient-binding compounds known as anti-nutrients, including phytic acid and lectins. These bind to essential nutrients and prevent absorption by the body. So, even while a diet may be high in nutrients, you may still end up deficient, because your body can't absorb the nutrients.

Third, many women with PCOS also do better on a lower carb dietary lifestyle, though this may be a double-edged sword. Many sources on thyroid issues say that low carb lowers thyroid function (which is partly true, it does lower the amount of TSH the body produces, because one of the roles of TSH is carbohydrate processing, so naturally, fewer carbs = less TSH). However, Stop The Thyroid Madness, a site dedicated to dispelling the myths of thyroid issues (I highly recommend perusing the site as a whole), states that the most successful hypothyroid people are, in fact, low carb, with a net carb level of 40-50g (net carbs = total carbs - fiber) - http://www.stopthethyroidmadness.com/hypothyroid-and-weight-issues/ .

Now, that said, you may not have to give up being vegan, though cutting out grains and legumes will likely be a difficult task, since they're both found in pretty much everything vegan that isn't an actual fruit or vegetable. It might be a good idea to concede to having something like eggs and/or fish a couple of times a month to ensure you're getting the nutrients not readily available to you otherwise (and keep in mind, that you can find such animal products that were raised in humane and sustainable fashion that is more in line with your ethics). If nothing else, I think it may be worth taking a good, hard look at your diet and making sure that what you're doing is best for your health.

As for doctors, I recommend to keep looking for a doctor that actually understands the issues you're faced with. Getting prescribed HBC in the face of a PCOS diagnosis is pretty much the height of ignorance of the matter, because HBC very often causes more harm than good, by overloading the system with one hormone or another (usually estrogen, which is already usually high in our system). I recommend looking into holistic and functional medicine type doctors, to find someone that understands the role of diet in treating both PCOS and Hashi's, and doesn't just say "here's some HBC, good luck."

You don't have to go on BC (I don't plan on it ever again) but you must treat your PCOS in some fashion, because not doing so can and probably will lead serious healthy issues, most notably diabetes, but also a couple of types of cancers. It is really no joke.

Are you on metformin? It's certainly a better option that BC in my (obviously non-medical) opinion. Of course if you are not on BC and don't want kids then you'll have to ensure that you prevent pregnancy some other way (metformin is often used by women who are TRYING to get pregnant - but that is because it can make their bodies act more "normal", which is good for anyone really)

There are also herbal and natural supplements that have been known to help, so you can try working with a naturopath if your doctor isn't willing to try alternatives (and if you can afford it or it's covered under your health care plan).

Also diet is really important. I understand that being a vegan is probably important to you, but it's really important to understand that women with PCOS especially need complete proteins in their diet (and these unfortunately really only come from animal products). If you cannot eat these, you will need to work with a nutritionist to ensure that you are getting enough protein, and the right proteins with all of the amino acids. You want to really cut out grains as much as possible, and cut out sugar completely, so being vegan is going to make this a lot more difficult for you, but I'm sure it's not impossible. You just need to do a lot of research on PCOS and veganism. Diet is one of the biggest things for PCOS and because you already have restrictions on your diet, it's going to be more work for you to ensure you're getting enough of what you need.