First, let's talk about your vertigo issue. Vertigo means you lose your balance and can fall down during spells of vertigo. It can be triggered by some types of exercise, but in many cases it comes seemingly at random times. Sometimes it lasts just a few seconds. Other times it can be ongoing for hours or even days. When that happens, you pretty much can't do anything.
So vertigo can mean that you have issues with pretty much any martial art, standing or not.
What you might not realize is that the more head movement you do in the vertical plane (going from being upright to being bent over), will increase the likelihood of having an episode of vertigo. And so, I think that rules out ground-based grappling arts such as Judo, BJJ, wrestling, Sambo, and so on.
So above all else with vertigo, you want to minimize your head movement, particularly in the vertical plane. That means that styles like Capoeira, wushu kung-fu, and Taekwondo are probably out. They do a lot of spinning techniques like the tornado kick, and they also do some quick, bending over techniques like the butterfly kick. Capoeira spends a lot of time going from high to low and back again.
You've also stated that grappling arts like BJJ and Judo are too hard on your body. I think what you're getting at is that you don't want to have to struggle too much, exhausting yourself against an opponent who's grappling with you. That requires a lot of strenuous, muscle fatiguing exercise. And that's not what you want at your age. Fine for when you were younger, but now it's just not what you're looking for.
You indicated a preference for weapons based styles. Let me focus on that. For weapons, you might consider one of the following: Filipino martial arts (escrima / kali), silat, kung-fu, kendo, fencing, archery, kobudo, classical jujitsu (samurai ryu with weapons training), and bujinkan ninjutsu.
Of those, I think FMA (escrima / kali) might be the right fit for you. Yes, you'll be standing, so your vertigo can kick in at any moment. But all of them are standing, so you're going to run into the same issues with any of them.
As for head movement, FMA doesn't actually have a lot of head movement in general. There's a lot less than many other styles. At advanced levels, you can see FMA people doing "level changes" (essentially going from completely standing to almost squatting), but it's usually while keeping the head upright, not bending over. I think this is generally favorable for vertigo, or at least more favorable than most martial arts that employ level changes.
FMA starts you off with weapons from day one, usually with a pair of escrima. The techniques you learn will work interchangeably with knives, machete, and unarmed combat. That makes it fairly useful and realistic for modern self-defense, particularly for older people who can't rely on their muscles and athleticism like they used to. Yes, you will use your arms a lot in FMA, but it's nothing like BJJ or wrestling.
FMA systems are generally designed to give you quick, linear progression. There are generally no belts or rank. This can be very appealing to someone who's older. You don't care about rank, and you just want someone to bring you up to speed on something really quickly without a lot of BS.
In about 3 to 6 months of training 2 to 3 days a week, you'll be able to defend yourself quite well with a knife on the street. At least against unarmed assailants. The rest of your time after that will be mostly preparing you for fighting people who are armed with weapons also. That takes the most time to master. Sparring with practice knives (rubber) or with escrima will show you how hard it is.
Something else you might be interested in would be either Wing Chun kung-fu or Southern Praying Mantis kung-fu. Both those styles have a very stable head that doesn't bob around a lot. The stances are meant to be more or less stationary, perfect for when you're in enclosed areas without a lot of room to move around. And they have weapons training. Of those two, my preference would be for Wing Chun, because SPM requires years of doing essentially one basic, solo form over and over again until your instructor says you're ready to move on. Whereas Wing Chun progresses you much more quickly.
As always, don't begin any new exercise regimen without first consulting your doctor.
Hope that helps.