When the British left India, Mysore (Karnataka) rulers agreed to instrument of accession by newly formed Indian government. However there were standstill agreements which are still leading to disputes, while also legally maintaining the princely properties with utmost care. India even retained the British Armed Forces and ranks. Who authorized state standstill agreements
Karnataka belonged in Bombay Province (part of Deccan, along with Maharashtra of today)...
The context of the question, the Kavery river dispute, was an issue between the State of Mysore and Travancore. The standstill agreements have no meaning in this regard. It was meant to be a temporary arrangement for legal purposes. It's of no consequence anymore.
When the state boundaries were redrawn, based on linguistics, Karnataka was carved from parts of Mysore state and Deccan states. This made the implementation of river agreement based on the original agreements very complex, as they were based on considerations of overlapping regions.
The original agreements between Mysore and Travancore as parts of British empire expired a long time ago, in 1974. There are no remaining agreements in force.
The Kavery river dispute reflects internal Indian politics now, and has no bearing to the British Raj or any agreements at that time.
Edit: The details about the dispute, which is the context of this question, on Wikipedia