Matt, sorry your calendar got changed. Life is like that, isn't it. First, what do you have... what date, and Day Hike or Overnight???
Unfortunately, it would be pretty difficult, if not impossible to switch permits with someone. Finding someone willing to swap for your date would be tough, and then it isn't really permitted by Inyo N.F. (I suppose if you asked nicely, they might agree to allow you to add the other party leader as an alternate leader on your permit, and vice versa on theirs -- because
only the original leader or an alternate leader is allowed to pick up a permit.)
Your other option is to release your unusable reservation to the system, and then to study the
Unused Whitney Permits pages, and find a time when there are usually permits available. Then, go for a walk-in permit (they're free). When you check the unused permits, notice how many were available, especially Mondays and Tuesdays. Even those numbers are low a few weeks in August, but if you can find time to go on those weekdays, you are pretty much assured of getting a permit. (I always point people to the week following July 4. Seems to always be spots available.)
And a last possibility: Set your browser's "home" page to hit recreation.gov for your target dates, so every time you go to the internet, you can glance at what is available. If someone releases their permits, you can reserve them. Here are the links and URL code to use. Adjust the dates to what you need:
Day Hikes: http://www.recreation.gov/permits//r/entranceDetails.do?arvdate=07/01/2012&contractCode=NRSO&parkId=72201&entranceId=315840&pGroupSize=1
Overnights: http://www.recreation.gov/permits//r/entranceDetails.do?arvdate=07/01/2012&contractCode=NRSO&parkId=72201&entranceId=315790&pGroupSize=1
Good luck!