I agree with caregiver222 that waiting another 3 weeks is far too long for you to wait for further testing.
Get on to your doctor and let him know that your condition is worsening and that you need an urgent referral to a specialist.
It may be a gland problem or other issue. If other glands are also swollen, ask your doctor to arrange for you to have a blood test to check for non Hodgin's Lymphoma.
Chase up the department and let them know that your condition has worsened. Book a very urgent appointment to see your doctor.
If you are not well enough to contact your doctor or the ENT get someone to do it for you.
You need very urgent attention and you need to let the medical profession know that your condition is worsening.
Let me know how you get on.
Best wishes.
Incidentally a three week appointment is unacceptable.
I see absolutely no reason for the prednisone, and feel it is contraindicated.
There are many possibilities. I suggest you find a good teaching hospital to be evaluated at.
If you are near Sloan-Kettering I would make an appointment for an evaluation.