Thanks!
Like you said, after doing sh LimeWireOSX.sh you end up with a blank icon, but this is a runnable file. Someone told me how to run a file from the terminal, and it works in this case, so I thought I'd share in case anyone else is looking to know: from the same directory as Limewire in the terminal, type
./Limewire
(that's period slash the name of the file)