I'm sorry my last post failed to solve your problem. It seems like this is a hardware issue because unchecking the box did not correct the problem. When did you purchase your MacBook Air? I have reason to believe that it is a MacBook Air affected by the recall. Recently Apple has recalled MacBook Air models because of faulty Flash storage drives. These models were sold between June 2012 and June 2013. If your system is affected, you should make a backup of your system using Time Machine or other backup methods as soon as possible, and visit an Apple Retail Store for service.
If your MacBook Air is not one of the ones that have been recalled, you might want to check out the article that I've provided a link to below about how to solve performance issues with Mavericks.
If these instructions don't work, you should next try to boot into safe mode. I have included a link to an Apple article explaining how to do so below.
You also might want to just do a fresh install of the whole operating system. I've included a link to instructions for that, too.
I hope I helped!
Sincerely,
Austin
MacBook Air Flash Storage Drive Replacement Program
http://www.apple.com/support/macbookair-flashdrive/
Mac Basics: Time Machine backs up your Mac
http://support.apple.com/kb/ht1427
OS X: What is Safe Boot, Safe Mode?
http://support.apple.com/kb/HT1564?viewlocale=en_US
OS X: About OS X Recovery
http://support.apple.com/kb/ht4718