My answer touches on a lot of what everyone else has mentioned but I wrote it up anyway.
I think of any emotion like an alarm going off, especially negative ones. This helps focus on solutions first, but leaves room for expression. Positive ones tell me "potentially, stay with this", negative ones say "problem".
If it's at a point that you really must think about it, a quick test is to think fast (actually fast, a few seconds... against the grain, I know) for the most simple reason for the feeling. Might be off, might not be, but it's worth noting. Whatever the case, some stimulus has set it off. That could be good information, or it could be as "external" cause-less as a stroke or as "meaningless" as indigestion (to reference one not-so-paragon of touchy-feeling-ness) ... we are all one brain plaque or bad stalk of asparagus away from a very cranky day.
Of course, really, if we think about it again, strokes or indigestion itself is not meaningless, and nothing is without cause. It's something really happening to you, and it's worth following up on.
Anyway, much like that example where the answer is food-related, the sadness "answer" if there is one is going to be "useful" if you can find out the various discrete reasons why. That way you can see if there is anything to do about them, if so do it, if not now you have context for the feeling which is nice, I guess.
Given you said "Fi demon" I'm assuming you're an ISTP or INTP, therefore inferior Fe, and I'd suspect you might also benefit from a "problem solving" and context focused (feelings aren't special, welling up from "inside" some nonexistant but felt-sense of self separate from the world, they're a reaction you have to the environment) and dare I say, social, approach to feelings.
So you have a sadness. Sadness to me usually breaks down into 2 categories -
1) Discrete trouble. Some mess or hurt I can identify. My cat died. I realized I haven't talked to someone I think about often, who I was friends with for 10 years, for 3 years now. Just by accident and distance. Someone I thought would be good to work with is overbearing and so it's like mourning the person who could've been (but never was). I remembered I STILL haven't finished projects I promised I would do a long time ago, and must face the fact that they're not happening and if they did it wouldn't even matter. I realize I will die.
2) Generalized sadness because some concatenation of the above is happening at once and I don't like to fuck with keeping track of my own drama, "let it all wash away" so all I get is the grimy residue built up, like scale on the drain. This one is trickier because there probably are multiple present no-solution sadnesses involved plus just likely sleep deprivation or being stuck in a rut or (conversely) being over-exposed and nervy over it.
So, solutions. Any discrete causes of sadness, if identified, might be solvable.
Some will be very direct, rational solutions like, removing yourself from the bad environment once you notice you're in one, or taking that risk quitting a job you really want to take but keep getting talked out of, or exercising more because you realize barely moving probably generates sad.
Others will be "indirect" kind of irrational-seeming "solutions". These are the by-nature or for-now intractable problems, so the only thing left to do is mitigate, manage, self-soothe, and keep an eye out for any change in their intractable nature... because there's nothing else to do for now.
The issue of you dying some day, for example, and having to live with that knowledge. If you want to go into senescence research, more power to you, but otherwise it's going to be mitigating and getting by.
Some various ideas about the death one, given we all face that. One is the social mitigation.
Find some way, one you enjoy and doesn't bring any kind of harm, to express about it and connect with the related expressions of others. Photography or poetry for some reason come to mind. I'm more of a photographing decaying things person, myself.
It also creates opportunities you otherwise can't just waltz into to get to talk (awkwardly) about these intractable sadnesses. Which sounds horrible but does quiet the "alarm" going off about them. Your post sounds like you feel really weird about talking about being sad. I second this. You also note unlike expressing joy, there is next to no reward socially for repeatedly or intensely expressing sadness bluntly.
Even at designated-sad events like funerals there's always some people who seem to try to punish you for "messy" expressions. And if you're like me, once you do express, it is often messy and you wish you could take it back. Sometimes it's worth just being expressive about those negative things, messy or no.
But there is that other path, which is especially useful for social expression of sadness that lingers - art. Art mediates the mess. People who would look sideways at you for always saying something off the wall or inarticulately negative like "I hate life" over and over, or crying every day or becoming violent will not judge you, and may even praise or pay you, for sad photographs.
I can't entirely explain why that is, but I think it has something to do with sweetening the deal and helping them balance sharing your distress (aesthetics are nice even if the subject matter isn't entirely), and perhaps most importantly, signalling to people that this is an expression and not a demand they work every bit of this possibly intractable problem out for me right away. Art tends to invite other people's feelings too in rather than defining them for them or just attend to your feeling alone, unlike making more blunt pronouncements. Rather, it invites everyone to explore it with you.
However, this didn't work initially because generally I don't notice I have some emotion to express, and so go pick up a camera. Unlike the tried failure method of breaking down randomly, it requires a habit. So pick something that works and make it a habit. The stress and sad busting expression through the medium will work its way into it for you, I hope. And again it can provide context that isn't so negative to people to talk about the negative thing, whether cruelty or death or feeling trapped or over-exposed.
This can also be an effective sort of solution if what you're sad about is a near-intractable or long-standing social problem. Expressing and exploring, in arts or philosophy and rhetoric, ends up moving people if you do it well and so stands a chance of doing more than "just" making you feel better.
If you're sad about problems much bigger than your capacity to solve, giving yourself credit for helping toward a future, even long-distant future, solution to any problem like that is good too. Just because you can't eliminate all illness doesn't mean you can't be glad you put your time into studying just one, and just because you can't remedy all of society doesn't mean tidying up the area or helping out neighbors is meaningless. It means something in the moment, in the experiences of you and anyone else involved, and might just add up to some larger positive change as well.
One more active, kind of rational thing to do about death and its attendant sadness is making sure you identify what all you want in life, re-evaluate that semi-regularly, and as-consistently-as-possible go for it. So you feel like you aren't "wasting" what little we have. So you are less likely to have unforced errors, unforced regrets, and more happy moments that rise up out of paying attention to what you love and putting yourself in the midst of it.
If you notice you aren't meeting your definition of good life, think a bit, check the definition, and if it holds, pick one area to change and do that. Easier said than done, but still. Feelings as alarm, heed it, look for something to do about it, learn to turn it off (but don't repeatedly snooze or break the clock) in a way that best helps you live well.
A good time to note - get a normal amount of regular sleep and add extra grooming (ex: extra teeth brushing, cut your toe-nails), care-taking (ex: watering plants and yourself), or change-of-scene and physical (daily walk uphill both ways) type routines into life if sad. Yes this can be difficult to do but if you manage to do even just one it usually helps, no matter what the sadness causes are.
