Photoshop is really the wrong software for that job, in my opinion. Even frame by frame animation is a chore in Photoshop compared to most other 2d animation apps. I feel excluding the use of other software is shooting yourself in the foot here, and just making life more difficult that it needs to be. As has been mentioned before me After Effects would be a really simple, quick, and efficient solution.
But anything can be animated by manually drawing the effect frame by frame.
The solution in Photoshop is that you will need a VIDEO layer to be able to draw frame by frame – regular layers will not work, and will only, as you have discovered, copy the same content across the entire animation. When you look up the help for animating layers in the manual it does not seem to mention this directly – because it assumes you would want to animate a layer object, i.e. repositioning it (tweening) or animate the opacity.
1) create a new file (and make sure to create a VIDEO timeline, not a frame by frame one!)
2) go to LAYER–>VIDEO LAYERS–>NEW BLANK VIDEO LAYER
3) now start animating frame by frame.
To create that line animation I would probably draw the entire line first and duplicate that to all frames. Then us the eraser to remove the line step by step.
This tutorial will also help: https://vimeo.com/23707998
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