So somehow Kagome turns into a dog demon, goes to the east, kills dragons and becomes the mother of dragons, and Sesshomaru has been following her because she’s apparently using this demon etiquette that intrigues him. He falls in love with her (in a relatively short amount of time) and they become mates. The last scene is that Kikyo forces Kagome down the well and Kagome becomes the new daughter of the dragon lord that killed Inu no Taisho. I read it here on Dokuga,

Hello!

The story you are looking for is called If Wishes Were Inu by Corvus Draconis

This story is rated MA so no YIMS!