Are they monitoring your thyroid levels regularly to make sure you are on the correct dosage .If you have only recently been diagnosed it takes a while for them to get the correct dosage .Are you on levothyroxine ?
Anyone who has to take thyroid medication does not have to pay for their prescriptions as it is a lifelong condition.
Get your GP to have look at this gland if it is swelling as you can ...note ..can ..get a goitre with some thyroid conditions .none of this is life threatening.
Have a look at the support group.
http://www.thyroidsupportgroup.co.uk/
Don't mess around with medication ..take what the doctor has prescribed .If you are not happy go back and see him .
Good luck ..hope you soon feel better.