I have not heard a song (in Tamil) called Cauery Mahatmyam or Tula Kavery Mahatmyam.
Here is a link to Sanskrit prayer for Kavery
http://sanskritdocuments.org/doc_devii/kAverIstuti.html?lang=sa
This link
http://sanskritdocuments.org/doc_devii/kAverIstuti.pdf gives more details (with commentary)
Sri PR Kannan has written a summary in a small nook titled Tula Cauvery Mahatmyam. In chapter 8
of that book (page 30), he has given the 4 slokas (as in the sanskrit prayer) - I am pasting it below:
मरुद्वृधे महाभागे महादेवी मनोहरे ।
सर्वाभीष्टप्रदे देवि स्नास्यतां पुण्य-वर्धनी ॥
सर्वपाप-क्षयकरी मम पापं विनाशय ।
कवेर-कन्ये कावेरी समुद्र-महिषी प्रिये ॥
प्रदेहि भुक्ति-मुक्ति त्वं सर्वतीर्थ-स्वरूपिणी ।
सिन्धु-वर्ये दयासिंधो मामुद्धर भवांबुधेः ॥
श्रियं देहि सुतं देहि स्त्रियं देहि ददस्व गाः ।
आयुष्यं देहि चारोग्यं ऋणमुक्तं कुरुष्व माम् ॥
This link
http://blessedpair.blogspot.com/2012/01/three-rangas.html gives three Rangas in Cauvery
Srirangapatna - Shivanasamudra - Srirangam
Here is a rendition of Ranganatha Stotram
https://www.youtube.com/watch?v=vWvlUESkLHA
Shri Velukkudi has given discourses on Cauvery Mahatmyam - which you can search easily
Hope these are useful to you
