Hello and welcome to MedHelp's forums. We're sorry you are having this difficulty and appreciate your question. As always, please work closely with your own physician for medical care and guidance. Typically, medication is not used in a goiter and this may be why they would not go along with your assessment of need for it. It can lead to hyperthyroidism and other issues. Wait and see is actually the most typical course of action. Non toxic multinodular goiters are usually slow growing. However, you now have what appears to be 'compression' symptoms. As you do not have thyroid symptoms, that puts treatment in the area of surgery. How extensive is based on each patients needs. They are not suspecting malignancy and it has been slow growing, so it is a matter of controlling your symptoms if caused by the goiter. They could choose to do a lobectomy or total thyroidectomy. That they are choosing not to do the surgery is something that you can choose to seek a second opinion. Here is information regarding treatment of non toxic multinodular goiters http://endocrinediseases.org/thyroid/goiter_treatment.shtml.
There are other reasons for dysphagia or difficulty swallowing. https://www.webmd.com/digestive-disorders/swallowing-problems#1. Acid reflux is one thing to determine if you suffer and the symptoms are not always as you think they may be.
This sounds very frustrating and I'm sorry you are left just trying to get help. Always know that at some point, doctors will listen so keep asking. Are you seeing an endocrinologist and this is who is still giving you a treatment plan of the goiter to just watch?
Request a biopsy via their website/patient portal so that it is documented, then see if they move forward more aggressively.
There is a news host, I think her name is Deborah Norville, who just had a biopsy of a thyroid mass she had for a long time. Google her story.
I underwent two thyroid biopsies. Not all nodules require biopsy, they can tell by ultrasound if there are concerning questions about the nodule to help determine if a biopsy is necessary.