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Have you tried copying the moving text to another layer? If you move the frames along a little it will start later than the original one.
Also try converting the moving text frame to a movie clip. You can then add numerous instances of it on different layers.

Select all the frames you want in the clip, then hit Insert New Symbol, and choose Movie Clip. You can then go back to the main stage and delete those frames, the movie clip containing them will be in the Library. Select the layer you want it on and drag it onto the stage.