Sure, it's possible to sell someone your copy of LSX, you just have to include the dongle in the sale.
To help with audio syncing, download audacity and open the song files in it. There you can align the audio (if they are the same recordings) without too much trouble. Usually there will be an extra bit of silence at the beginning or end.
If you can't find the right song, it's possible to re-synch by moving events. To move many events at once, you can select them by clicking on them (or drawing a box around them with shift+drag) and then they can be moved with the mouse or keyboard arrow keys. You can use this with the audio spectrum visualization to aid in fast alignment.