Nodules can cause hoarseness, as can hypothyroidism. Have your thyroid hormone levels been tested? If so, what were your actual levels of Free T4 and Free T3? Please post reference ranges with any lab results as ranges vary lab to lab and have to come from your own report(s).
In addition to hypothyroidism, nodules can press on the vocal cords and they can't always tell that unless they've actually gone and looked. Don't let them just say "the nodule isn't large enough" because it doesn't take a large nodule to cause trouble.
You always have the option of getting a second opinion and should exercise that option whenever you don't feel comfortable.
The head/neck MRIs might show something, but if you've had thyroid blood labs, we could start there for now.
How long ago did you have those thyroid levels tested? Were the T4 and T3 Free T4 and Free T3 or are they Total T4 and Total T3? If it doesn't specify on the lab report they will be Total T4 and Total T3, which are considered obsolete. Ask your doctor to test Free T4 and Free T3...
Sometimes it works to be the greasy wheel; sometimes not.
You could have acid reflux; that's a classic symptom of hypothyroidism. What treatment was recommended?
Free T4 and Free T3 were far from optimal when you had the tests done in Feb. Most of us find that we feel best with Free T4 about the midrange point and Free T3 in the upper half to upper third of its range... Your Free T4 is only at 30% of its range and your Free T3 is only at 48% of range. In my opinion, you could well be having hypo symptoms even though your levels are well within the normal ranges. This is not unusual at all.
The problem is going to be getting a doctor to start any kind of treatment, especially. with your TSH where it is.
You can try talking to your doctor about this, though and see if you get anywhere.