If I remember correctly, I think some of the secondaries explicitly ask you to type. I'm not sure if handwriting would be detrimental necessarily, but I think the effort to use a typewriter is worth it. The way I see it, schools get thousands of applications for 100+ spots on average. Handwriting an application could be seen as being unprofessional, and could possibly get your application tossed for that reason alone. Why risk it, when you've invested so much more time and money into the situation.
As for using a typewriter -- most libraries have one. Are you still in school -- perhaps you can talk to your advising office and they can offer you one to use. When all else fails, you can usually go to a Kinko's and they have typewriters available -- sometimes they charge, sometimes they don't. Or you can ask around your friends and family -- there's probably a good chance that someone you know has access to an office with a typewriter. If you look around a little, it shouldn't be too hard to find.