Yes, it would be appropriate for you to see a medical provider so that you can have your throat properly examined. A sore throat, primarily in the mornings wiht improvement in the afternoon, may be due to post-nasal drainage at night, accompanied by mouth-breathing. Allergies can cause this, but so can back to back viral infections. Mononucleosis is a virus that can cause a sore throat. People also worry about a lingering Strep infection, so this can be tested for during your examination if your medical provider feels it is necessary.
As you have read, there are many possibilities to consider, so you are wise to seek medical care at this point.
I'd say so. It may be something minor, like dryness due to furnace season being upon us or post-nasal drip, but since the symptoms are bugging you and aren't responding to over-the-counter (OTC) remedies, you won't lose anything by seeing the doctor. It could be you have a sinus infection or something else that's treatable. And then you won't be uncomfortable anymore.
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