I have horrid handwriting too and out of all the jobs I've applied for, only one of them required me to handwrite something - it was for a social trainer position working with the intellectually disabled, where they wanted to see on-the-spot proof of my English competency. Otherwise, every application has been done on the computer and my resume's also on my computer. I usually just write out cover letters in Open Office and attach that and a copy of my resume to an email.
If you have to personally hand a resume over, just print one off from the computer. Computers have made life much easier in that regard.
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Said the apple to the orange,
"Oh, I wanted you to come
Close to me and
Kiss me to the core."
Think you're ASD? Get thee to a professional!