You've got really hard water, and the white dust is calcium. I own my home and have city water, and therefore have a huge water softener in my basement that all my water is run through, and I have to add salt pellets to it once a month. Honestly though, even with the huge water softener machine I still have white calcium deposits all over my faucets, dishes, everything. So you need to do what I did, as I too have to use a humidifier, and if I just put tap water through my humidifier, even after using the water softener, I'd have white calcium deposits everywhere. So I decided to take another approach: STOP TRYING TO GET RID OF THE CALCIUM DEPOSITS AT THE HUMIDIFIER, IT WON'T HAPPEN AND YOU'LL SPEND A FORTUNE TRYING! YOU NEED TO SOLVE THE PROBLEM WITH THE WATER YOU'RE PUTTING INTO THE HUMIDIFIER, THIS IS THE ONLY WAY TO STOP IT!
I'm kind of a professional on this topic because I had to be out of force... I've had chronic kidney stones since I was 18 years old, I've had surgeries to remove them, and I've probably passed around 75-100 stones that caused enough problems and horrible pain that I was aware of them. Many a night in the ER hooked up to pain meds, fluids, and antibiotics...uhg. So when I look at the water coming out of my humidifier and I see it dry, this is how I gauge what not only I'm drinking but what my birds and dogs are drinking. So far, after installing a water filter that is specifically made to filter out hard water on every faucet in my house, I've had no more white dust in my house, or any visible in my dog's or my bird's bowls.
I have vehicles and I drive, so I actually go to Walmart once a month and buy 15 gallon-sized jugs of purified drinking water for my birds and dogs. If you go to the soda aisle in any Walmart they have one side of the aisle that is nothing but gallons of their brand (Great Value) of spring water, purified water (what I buy), and distilled water (just as good). They cost $0.88 a gallon, so for $13.20 a month I don't have to worry about my pets. Now 15 gallons covers all the drinking water as well as any water I might use to mix or make food for the birds, for an Australian Cattle Dog, a Shar Pei, a green cheek conure, a Quaker parrot, a cockatiel, a Senegal parrot, 7 pet budgies, and 8 breeder budgies. So obviously if you only have one bird then you could probably get away with 5 gallons a month easily, and they actually sell a 5-gallon container of purified water for $4.00 I believe. So whenever you go for groceries every month you could just buy 5 gallons of purified water for Jasper's drinking water for the month, plus another 5 gallons to run in the humidifier, all for under $10 a month. That's a hell of a lot cheaper than changing filters in a humidifier every month, and it's likely changing that filter does little if anything for hard water. So this is option #1, and it's actually a cheap, safe solution for Jasper's drinking water and for completely keeping the white calcium deposits out of your house. Option #2 is buying water faucet filter specifically made to eliminate hard water. This is much more expensive than just buying purified water for your humidifier every month, but it will make your drinking water safe for you, Jasper, and everyone else, as well as totally eliminating any calcium deposits coming out of your humidifier and then into your house.
I bought the first one at Lowes I believe, and if I remember I paid a fortune for it, but it was immediately noticeable that I had no white junk coming out of or collecting in/on the humidifier after I added tap water that went through the filter to it. So I then got online and bought the same filter for both my bathroom sink faucets (I put the first one on my kitchen sink faucet). I believe I bought them on EBay, same filter I got at Lowes, and I got them for half the price. Where they get you is you then have to change the filter elements every month in these as well, and they aren't cheap, though they don't cost as much as humidifier filters, I believe I pay around $10 a filter. Do a Google search for "hard water faucet filter" and you'll get multiple to choose from.
Honestly, if I was in your situation (renting and mostly concerned about what's coming out of the humidifier into your apartment), I would stop buying humidifiers and filters for them, because you're never going to find one that will eliminate the calcium deposits that result from you putting hard water into the humidifier. It's virtually impossible to eliminate hard water without chemically treating it with sodium and changing it at a molecular level. If I were you I would just buy gallon jugs of purified water every month and only run this purified water through whichever humidifier you like the best. At $0.88 a gallon for purified water at Walmart you can't beat it. I know you said you don't have a car but you must go and get groceries monthly somehow, so I would just add the purified water to your monthly shopping list and be done with it. I don't know how often you run your humidifier, but even if you had to buy 10 gallons of purified water it would only cost you $8.88 at Walmart, and you would probably have enough to also give this to Jasper as his drinking water too.
"Dance Like Nobody's Watching"
