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That's IT!. After I read your post I went and inspected the situation and sure enough, the A string goes in a almost a straight line to tuner shaft. I picked up the bass and pressed my finger down on the A string on the headstock side of the nut and pluck the string. All the buzz was gone! I loosened the string and pressed down pretty hard. That fixed a good deal of it, then I used a wood dowel to press the string further into the nut and that fixed the rest of it. . .
. . . well at least for the most part. There is still a tiny bit of buzz when it's unplugged, as as soon as I fire up my amp the buzz can't be heard even at low volume. (I used to be able to hear LOUD buzz, plus I could feel it rattling the bass as I played)
The next time I change strings I'm going to add more wraps around the shaft to help sharpen the angle.
It feels like I have my bass backThanks for that!
(Pic from before you told me what they problem was)View attachment 7125186
That's great!
More wraps will definitely help a lot and probably get rid of the buzzing. You can probably add an extra wrap or two on the E-string as well, for better tone on your open E.
A bar-style retainer that includes the A-string will be even better. Not only will you get rid of the buzzing/rattling, you will also get a better tone from your open A-string, due to higher pressure over the nut. Ideally you would want all the strings to leave the nut at more or less the same angle as the D+G strings in your pic. I don't think you will get quite there with just more wraps.
But of course, bass players have gotten excellent results with Fender basses without an A-string retainer for DECADES. I'm not saying you NEED a retainer for the A-string. This is just to get an even better tone out of the somewhat compromised open A that can be problematic on most Fender basses.