Well, then it still sounds like it could be the screen, but there is a possibility thyat the problem is the digital main board, and there's really no easy way for you to tell.
There are basically 3 things in an LCD TV: the Power Supply, the Digital Main circuit board, and the actual LCD screen itself. The power supply isn't suspect because the set comes on, responds to commands, and otherwise works.
But the Digital Main and the screen have elaborate handshaking and communication that has to continually happen, and if one or the other interrupts this, then there are all sorts of picture problems that can happen. But that's also why I asked about the TV's menu, because if it looks OK, then the screen is obviously not at fault.
But if it has to warm to start working, the backlight in the screen will warm up the screen the longer it's on, so that makes the screen suspect, but there are also components on the Digital Main that get normally warm, too, so that complicates it.
If you are comfortable opening the set, I do have one test that may isolate this to the screen or the DM, and it involves "circuit cooling" spray (I think you can get this at any Radio Shack, but I can also tell you where to order it on line). And what you do is simply wait for the set to start working, turn it off, then spray this coolant on the DM board, turn it on, and see if the set comes right back on (because the screen is still warm) or if it's back to lines.
Let me know if you'd like to try that.
Otherwise, of course, you'd have to get a tech involved.