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No. You should continue the prednisone at the prescribed dosage until it is time to step down from the medication. Discontinuing use of the steroid can cause problem if you don't step down gradually from it.
Prednisone is a steroid and it DOES cause increased thirst and urination. This is normal. The symptoms will improve with time or when you stop the meds.
Let me know if you have more questions about this.
I am surprised to hear that the vet doesn't want to taper them...
You might want to give them a call and ask if there would be fewer side effects when you stop the dosage if you taper them off.
I would continue using them for the rest of the prescription. They do give the dog a lot of relief from the cough.