What fuel are you burning?
The reason I ask is because my late mother had what seems like the same problem, and it was found to be caused by the timber she was burning being too fresh and it gave off a gluey residue during and after burning. Eventually the stuff would go on fire with the least amount of flame present in the stove but for the life of me I cannot remember how it was solved, if at all. Hopefully, it's the wrong answer i'm giving you and your problem will be an easier one to work out.