Dry cough with tickle in back of throat for 3 weeks now. any advice for treating this?

Post nasal drip.: Most likely post nasal drip. But need to r/o sinus infection versus other causes of cough. Otherwise you could used nasal saline twice a day.

Dry cough: Dry cough is unlikely to be postnasal drip as the mucous would be coughed up, so the cough would be productive. Consider reflux, asthma, a viral syndrome (unlikely to last three weeks, though). See your doctor for an evaluation.