Welcome to the forum!
This may sound crazy, but consider replacing the battery if it hasn't been changed in quite some time. The LS is pretty finicky about the battery being in very good condition. Even if a load tester shows that the battery is fine, it may still not be sufficient to keep the LS electrical system happy.
You can use the "Search this forum" feature from the LS forum main page to see if others have had similar issues solved by replacing the battery.
Otherwise, you're going to have to use a datalogger or scan tool with datastream mode to monitor some of the fuel and ignition system values in real time to determine if you have a fuel issue or ignition issue.
Oh, when's the last time the fuel filter was replaced? Also, have you checked the air filter and cleaned the Mass Air Flow (MAF) sensor?
-Rod