Replacing packing in the water is no big deal. If the nuts turn already, and the stud does not show obvious crevice corrosion, which is likely wha caused the nut to disintegrate, again no problem. Get a wire brush in there and see if the crud cleans off. I don’t see a completely obvious crevice issue on the stud, think it can be serviced. No way I’d add packing to an old crusty gland with almost certainly hardened packing. People worried about maintaining these things is how they get old like this.
Buy a couple corkscrews or three, cause if the packing is ancient, good chance you will break at least one. Get that old packing out of there. Glands should not drip at rest. It won’t “blow out”. In fact, you will have to wrestle to get it out. Not a bad idea to have a cheap mirror to ensure you got all of it too. Worst case scenario, you can stuff a rag or even paper towels in there and stop any flow and go shopping for hours. Usually with the entire packing removed, you will laugh at how little flow there really is and understand why “blowing out” is just not a real concern. It’s completely manageable unless your bilge pump is in a dangerous condition, if so, fix that.
I can’t really say whether that shaft is in bad shape, the can develop a bunch of crud and that’s all it is. I’m worried about the shaft being severely scored under that packing. If it is, it will never really seal right but better to get fresh packing in for now anyway. Also, if the alignment was off, prior owner almost certainly overtightened in false attempt to solve and packing will be like a rock.
Got to stop telling people to tighten glands as the primary means of stopping dripping if they don’t know the condition of the packing and alignment. Causes more problems than it solves.