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It might well be that the truss rod is a bit too tight, and that is better news than if it was not tight enough and could not be tightened further.
Since it is easier and safer to adjust the heights of the bridge saddles than to turn a truss rod, this may be what the last owner has done, or someone on their behalf.
Raising the saddles will mainly stop buzz on the higher frets, but can leave some on the lower ones.
What you noticed might even be the reason that the bass was for sale, who knows ?
Anyway,
IF it had been adjusted as I just speculated, then loosening the truss rod ought to help your issue, and as you
gradually do so, perhaps a quarter turn at a time, a day or so apart, you should notice that it does, but your action will also get higher overall.
THEN, you can begin to
gradually lower the bridge saddles, and that will most likely make up for the higher action, but your buzz will hopefully be gone, and if you keep going sufficiently far, it might move to the higher frets.
You might be able to speed up the process a bit by pushing the headstock upward as you loosen it, and because you are loosening, the strings can remain at normal tension throughout the entire time.
ONE OTHER THING : STRINGS [and you might wish to examine them
before turning the truss rod]
What make, type, and gauges are they ?
If they are a set with a 4th which is 90 or 95 thou, try a set where the 4th is 100 thou or more.
I like the LaBella 760FL DT Flats series [43-60-82-104], and use them on other short scale basses, so 760FL-S, but if your Mustang bass has strings which pass thru the body at the bridge, they make a special 'MUS' set for it, otherwise you would just need the short scale version of 760FL which I use.
Short scale basses frequently begin or end up with strings which ought not to be on them [too light gauge], because some string makers incorrectly and stupidly insist upon only offering such strings for short scale basses, and far too many players blindly trust them.
Going heavier, with any brand which offers them, might help pull the neck up enough without adjusting the truss rod, and then you can just lower your bridge saddles a bit if this succeeds.
Hopefully one or both of the above solutions will be of help.

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