I'm sorry you are going through this. I had something similar happen to me in 2018, except my nodules were slightly bigger than yours, and I got the fine needle biopsy. I had a ton of hypothyroidism symptoms (which were undiagnosed up until the point that I realized that my thyroid was full of nodules and that many of the weird symptoms I had been experiencing for years were probably thyroid related).
I hated having anything touch my neck ever since I was an adult, so for 20 years prior to realizing I had a thyroid problem, I had neck sensitivity issues. I had my first ultrasound and was immediately referred to an ENT, who told me that it is completely normal for a woman my age (37 at that time) to have thyroid nodules (is it normal to have a thyroid full of 1 cm nodules? No, but the resident who was trying to convince me my hypothyroidism was all in my head did not tell me that). I was told I probably didn't have Hashimoto's, so I didn't go get the blood test I had begged the ENT to order for me until I had my fine needle biopsy and I was already scheduled for a hemithyroidectomy because one of my nodules was suspicious for thyroid cancer.
What can you do right now? Do what gimel recommended, and get tested for thyroid hormones and Hashimoto's. If you have Hashi's and you are experiencing hypothyroidism symptoms, taking supplemental thyroid hormone might help alleviate some of those symptoms. If your nodule that is 1.3 cm has any features that raise the risk that it could be thyroid cancer (hypoechoic, vascularization, part cystic, part solid, calcifications), you could try to get a second opinion and see if they recommend a biopsy, although most thyroid cancers are very slow growing, and the risk increases as they get bigger, so many hospitals/doctors offices use 2cm as the minimum size before biopsy. I had some nodules that were over 2cm, but the one that was hypoechoic and which caused the need for a hemithyroidectomy was 1.3 cm and hypoechoic (and suspicious for a neoplasm by biopsy).
When I had the rest of my thyroid removed, my throat immediately felt much better - I had no idea how much inflammation had been going on. My Hashimoto's had probably been going on for a very long time, and in addition to a very inflamed thyroid, my ENT pulled out 3 lymph nodes (first surgery), another 3 lymph nodes stuck to the side of my thyroid (surgery 2), and a giant clump of 8 lymph nodes (2nd surgery)- all the lymph nodes were about 1 cm in diameter, and were full of destroyed thyroid tissue (and thankfully no thyroid cancer) - my immune system was putting up a very aggressive attack against my thyroid, and clearly winning.
It is possible your throat issue is completely independent of Hashimoto's, but that would be one heck of a coincidence (if you have Hashimoto's).
I know if the thyroid/goiter is too big and causing pain or blocking food intake or airflow, they might remove the thyroid even if it is not cancer. In my opinion, as a person living with no thyroid, trying to avoid thyroid surgery and trying other options first that would allow you to keep your thyroid is what I would do, unless the pain/sensation is unbearable. The three years after my thyroid surgeries were the three hardest years of my life. (This is not normal for most people who have thyroidectomies, but it is something I recommend considering if given that choice.) It took me much longer than expected to get to a stable dose of thyroid hormone, complicated by the fact that I was also anemic at the time (undiagnosed - I believe my doctors were trying to help me get better, they were just not very good at times). I'm finally at a much better place, but with supplemental thyroid hormone as my only thyroid hormone I get, I'm constantly trying to optimize my energy levels. I've finally returned to full time work in a career that I love, but I get home from work and I'm much more tired than I used to be, even when I had some hypothyroidism symptoms but still had a (semi functional) thyroid.
So - at this point it sounds like your nodules might be too small for a biopsy, and they will keep doing ultrasounds at intervals until the nodules get bigger. (And keeping your thyroid is not a bad thing, in my opinion. If there is a legitimate reason to have it removed - do it, but otherwise I'd hang on to it for as long as possible.) If you do have hypothyroidism, supplemental thyroid hormone might help with some of your symptoms. If your doctors won't listen to you, find new doctors. It has taken me close to 4 years to start to feel like myself again. I wish certain things (like anemia) had been diagnosed much earlier, but I'm glad I didn't give up on myself. Keep advocating for yourself.
And, I don't think the symptoms are all in your head, and they are so similar to mine, and my doctors convinced me I didn't have Hashimoto's, that I didn't have hypothyroidism. The months leading up to thyroid surgery I had extreme anxiety and was blaming myself for destroying my thyroid - which of course I didn't purposefully tell my immune system to attack my thyroid. I don't think doctors realize the harm they can cause when they tell patients they are not experiencing symptoms they clearly know they are experiencing.
I cannot speak to your us results.
The first thing I would do, and right away, is to get tested for TSH, Free T4, Free T3. Along with those I suggest testing for the antibodies of Hashimoto's Thyroiditis: Tpo ab and Tg ab. If the doctor resists, insist on those, since you were told it might be Hashimoto's. If you have Hashi's there is definitely something that can be done to alleviate your symptoms.