I agree that the thyroid could be behind a lot of this. Hopefully your vet told you that the thyroid meds can be refilled at human pharmacies -- it's the same drug prescribed to humans. I think it's on the generic meds are on the $5 list at some pharmacies (like Walmart). You can check GoodRX.com for a list of prices in your zip code.
I would stop the oatmeal shampoo. Oats are a potential allergen. I would use a chlorhex veterinary shampoo instead -- like this one -- look at the Amazon reviews to see how well this type of product works for a lot of dogs with skin issues:
You can use it weekly or even 2x weekly because it doesn't suds up or strip coat oils like regular shampoo. It knocks down the microbes that tend to opportunistically attack inflamed skin, while also washing off contact allergens. That's important, as it can avert the need for antibiotics as skin infections tend to follow allergic flare-ups in many dogs.
I agree about trying Apoquel if you are at the point of giving the dog up. It gives many dogs instant relief. I think of it as a bridge to get them through a bad episode while you and the vet derm come up with a long-term plan.
Another atypical thing you can try if you're at the end of your rope is asking your vet team about trying a sulphur-lime dip -- it's not normally used for allergies (it's normally used for fungal infections, or mites). However, if her skin is a mess already, it may help with that -- it seems to help skin heal very fast and give some relief. It stinks to high heaven (rotten eggs) and stains porous surfaces (a disposable kiddie pool outside works pretty well for this). It's about as safe as anything you could try, even for puppies. It's kind of a "hail Mary," because of the smell and mess, but sulphur may have some skin benefits.
Grass allergies are increasingly common, and they're a work to manage (wiping paws, etc.). Food allergies are easier, once you find the right food.
Please think about what happens to her if you rehome. There are not many people out there looking to adopt a sick, expensive dog. I worry that she's not likely have a happy ending unless you love her enough to commit to finding a solution. I know it's frustrating -- I have a food allergy dog who's a very expensive keeper and had me feeling like a failure as an owner when he was sick and we didn't know what was causing it. Persistence saved my dog. Your dog needs your persistence right now. If you'll go all in to help her, you'll have a lot of support here.