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Chronic cough can be due to several things, and depending on other factors such as smoking history, weight loss etc may need a full evaluation.
If the problem is only in the morning then it is possible that your symptoms are related to GERD (due to irritation caused by stomach acid) and if that is the case then you need to continue with Prevacid, in fact possibly increase the dose.
However if this is a rhinitis problem then it is well recognized that the use of a steroid nasal spray can be helpful, If your doctor is not familiar with this then you should see an ENT specialist.
However here is a link that may be helpful: here
My only concern is that the spit is not clear, as that may possibly imply an infective cause and you may need some oral antibiotics if a sample provides evidence of infection.