Kodaikanal hill station would be the best for you.
many foreigners come to this place and be here for around a month or so for very reasonable prices.
do check out my trip and things to do in Kodaikanal
https://tripoto.page.link/2Xep
Try Zostel. Their hostel rates are very cheap. They do have their properties in most of the popular places and even in some off beat places. If snow fall and harsh winter wouldn't be a problem for you then you can try for Himalayan Regions where you will get great accommodation at rock bottom prices cause of winter.
I think u should visit coorg in karnataka, u will find good rooms at cheap rates, it would also depend which month are u going , depending upon the season , rates are flexible