I am not a times series expert, so I will hold off making a particular recommendation. We can wait and see if any one else chimes in to help.
To my knowledge you would use your coefficients to score future data (months). So perhaps you need to control for months as well. This way you can get values for month name as a categorical variable not just as an ever increasing continuous variable - which I believe would leave you with a straight line instead as a more volatile looking undulating line.