Though I understand your dilemma, I want to agree with what has been said about visits - don't go if you're already certain you will not accept the offer. In response to your other question, though: this depends somewhat on the kind of housing you're looking for (studio apt., family housing, etc.) and the school you are planning on attending. It is always better to hand in a housing application earlier rather than later - this will give you an edge over all the applications that will be submitted directly before April 15th. This matters even more at schools/in buildings where you must compete with undergraduates as well as graduates. Additionally, if you're really eager to get in to one particular housing building, I recommend choosing a relatively early "move-in date:" I've been told that this, too, gives your chances a boost because the vast majority of students are looking to move in during August.