After accidentally inhaling soda ash, my noise and throat have had a raw, burning feeling. There has been some improvement. Should I be concerned?

Soda ash is very… Irritating to respiratory tract, skin, eyes. If you haven't done so already, you should seek medical attention asap to prevent complications.
Soda ash. Soda ash (also washing soda - or sodium carbonate is a mild alkali agent so it can cause local skin irritation especially to mucus membranes of the nose and mouth/throat. It should be temporary. - drinking lots of water will help. For the nose you can try nasal irrigation - using a netti pot may help. Good luck.