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You can cut as many corners as you want, but the ideal first thing to do would be to find the source of the water damage and take measures to prevent it again. (It could range from drips, leaks, seeps, floods, or even a momentary spill that was not cleaned up.) Doing so might require removing the current vinyl floor, because the water could be seeping up through the floor.
As for leveling the floor, there are cement products on the market that can level concrete floors. They pour in like a soup, spread from wall to wall, and gravity smoothes it out into a nice, level surface. It then hardens to a nice, level, workable surface. They are reputed to do a great job. But you need to empty the basement or elevate everything in order to use them; otherwise, whatever you leave on the floor will become fixed into the floor.