Besides cold etc, Delhi has been unbearably polluted during the winters for sometime now, gas masks et al. Avoid it like the plague. (There is a case for doing so at any time of the year, actually)
In areas like Cochin, Varkala and Trivandrum it's common to find Europeans who come for the whole season.
But I'd strongly suggest not booking anyplace beforehand for a long stay. See what area you like then look around for a long-term place.
(I hear what you're saying about avoiding tourist areas, but you'll find the practical aspects of getting settled much easier, at least for your first winter, in an area accustomed to lots of seasonal foreign stayers.)