No, you are not at a serious disadvantage. I don’t know what percent of applicants don’t have a chance to visit campus, but it is considerable. I would even go so far as to say that a significant percentage of the applicants who consider Tulane to be in their top 3-4 have not visited. After all, over half the students every year are from the West Coast area and New England/Northeast. That’s a trip that many just cannot fit in before applying. A lot know they can only visit once and many opt to wait and see if they are accepted first, so as to maximize the utility of the trip. So obviously they had not visited before writing their essay either.
You say it is a school “in which (you) are highly interested”. There must be a reason for this. Focus on that. The most truly felt and honest reasons will come through the best. Also read this: http://tuadmissionjeff.blogspot.com/2013/09/the-optional-statment.html
Good luck!!