Converting a regression problem into a classification problem
I think what you should be looking for is ordinal regression/classification in order to utilize the ordering of the classes.

Example python/matlab implementations, with some extra resources.

For fast integration with your system, you could create a new ordinal regression model using the outputs of the already trained network as input to the new ordinal model and the true ordinal value as the output.

For another quick/dirty solution you could try simple Mean Absolute Error or Mean Squared Error, according to this study, between the predicted and target classes of your system, as it is right now, instead of the categorical cross-entropy error function.