If you want to stay for a month or more you can also consider seasonal lets of an apartment or a studio. There are several agencies in town, I think at least two of them run by British folks, e.g. http://chamserve.com/EN_Seasonal_Lets_Chamonix.htm
(had a good experience with them last year, friendly people, negotiate your own deal); Or check the billboards in the hostels, there are always postings for apartments for rent or roommates wanted.
Rates for a studio in town start I think somewhere around 450 a month (sometimes incl. utilities) but I think you'd need to rent for a few months to get their low rate. They even have apartments with bunk beds that you could share with others.
//Edit: Correction - March is still winter season during which rates are usually double, so you're probably best off with hostels or looking for room mates wanted postings.