It does sound like the truss rod simply needs tightening. Fret levelling won't fix this. Put the hex-key in the truss rod nut and turn it clockwise/to the right. It may take several full turns, but eventually the nut should start to provide resistance and the neck should start to lose it's convex bow (I assume the neck is rather like the bottom diagram at the moment).
If the truss rod nut doesn't tighten up after say 20 full turns, then get back to us, as the truss rod is probably defective.
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