It was only discomfort when wax was removed and after that no problem indicates your ear is fine.
It is usual for the patients to cough when wax is removed by instrument. Because a branch of nerve
supplies the ear canal
. Tickling of this can cause the patients to cough.
Following cold you developed sore throat
and pain on swallowing. This indicates secondary invasion by bacteria.
If your sore throat increases or persists you may need a course of antibiotic. If it is reducing no need.
It is not due to irritation from ear wax
removal. Because wax gets collected in external ear and it has no connection to nose or throat. Only middle ear has connection to back of nose.
You take plenty of warm fluids
Antiseptic lozenges to ease your soreness in throat
Pain killer if it causes pain disturbing your sleep
Wish you good health.