House sitting is a fickle game by its very nature. Finding a house sitting opportunity that approximately matches your requirements may be possible if you advertise for an assignment well in advance. However, you may need to work around the home owner's requirements by arranging some other accommodation at either end of the assignment.
If your need for accommodation is urgent I recommend using the website
http://www.sabbaticalhomes.com, which is very well linked into the global academic community. They describe themselves as '...the leading worldwide internet-based directory for home exchange, rental and tenant listings opportunities devoted to serving the academic community...providing a listing service to academics worldwide who temporarily need to offer or search accommodations for rent, exchange or house-sitting'.
While their emphasis is on home exchange and short-term rentals as well as house sitting you may need to exchange your home or pay for accommodation to find something suitable. Note that you must have an email address that is attached to an academic institution or be able to provide an academic reference to be able to use this service. Creating your own sitter available listing is free but should you find accommodation through the website (even if it's rented accommodation) you are expected to make a success-based contribution to the website of between US$30 and $70 depending on your circumstances.