I have been diagnosed with bells palsy. I am also experiencing a bad sore throat and trouble swallowing on the impacted side. Should I be alarmed?

No, may be unrelated. It is possible that this may be unrelated but it may also be a viral infection you had acquired earlier which might have triggered the bell's palsy. There are theories linking some viruses to the disease so it may be possible that you acquired the viral illness prior to the onset of the bell's palsy.