Clustering vs. Classification
Clustering is unsupervised which means that you do not know the classes and / or have no examples of correctly labeled texts.
Assuming you have some labeled texts then we are talking about a classification problem.
Your current approach of doing separate binary models for each label is very basic but still sound. What is your validation metric and score for each of the models? If the performance is already good then you might not need something else.
Another approach would be to train one model that predicts all classes and then to output the multiclass probability prediction. This would give you information about the best fitting classes but you would have to fit your loss metric accordingly, accuracy wouldn't make sense here for example.
Another approach would be to make a new factor variable which encodes all possible combinations of classes and train one model on this. However I suspect that the model would perform badly due to imbalanced cases and a high complexcity.