Our house has a radiant heating system in the concrete slab. Ever since I started running that more aggressively (it's on a solar collector) it hasn't been a problem. If you keep your books in an enclosed bookcase you might consider adding one of these:
https://www.amazon.com/dp/B00D1WYXR0/
I put one in the cabinet under our kitchen sink and it made a huge difference.
The theory of rust on tools is not simply a humid environment, but the changes in temperature that allow for the water vapor to condense on the metal, so you either treat the metal with a protectant (I use camellia oil) wiping down with a oil rag before storing away, reduce the humidity in a closed storage container (e.g., chest) with dessicant (the kind they use for drying flowers and sell on amazon works well), or use a "Goldenrod" which deals with both temp and humidity: https://www.amazon.com/GoldenRod-Dehumidifier-Installation-Operation-Prevention/dp/B00D1WYXR0?th=1
All are tried and true solutions. Shipwrights who deal with coastal humidity (and salt) have been known to use mutton tallow, I have a can of McQueens I got a long time ago from Lee Valley and put it on the planes in my traveling toolchest, very effective but you sort of have to wipe it off before using the tools. And a very small amount is all that is needed, a little can has lasted me over a decade.
The climate in Waimea has shifted a lot in the past 30 years according to the folks that did our solar panels; used to be in-town Waimea was a lot cloudier, now we average quite a bit more sun. It'll still be wet on the wet side, but maybe not as much as you're hearing from old stories.
Dehumidifiers will help for enclosed spaces, especially near ground level. For fully closed closets or cabinets I've had great results with these GoldenRod dehumidifiers:
https://www.amazon.com/gp/product/B00D1WYXR0
It's basically a low-wattage heater. Works miracles, totally maintenance free, but it can attract bugs because they like the warmth. We used rechargeable dessicant packs for a while. It was OK, but we didn't notice as much of a difference.
If you have one, run the heater from time to time to bake out the moisture in the house. We have a hot water circulation system in our floor, it runs off solar and gets good work done on sunny days. Airing out the house is also important, maximize those sunny days. If there's a plant shading the house or trapping moisture near it, maybe it needs a trim. Check the rain gutters too, make sure the runoff is deposited away from the foundation, makes a big difference.
Good airflow and targeted interventions work for us. We have a few cabinets with these (like under the kitchen sink), they make a world of difference:
https://www.amazon.com/gp/product/B00D1WYXR0
And yes, get rid of the carpet. If your house is shaded by large trees that will also have an effect, getting direct exposure to sunlight cures a lot of subtle airflow problems.
I put one of these in the cabinet under my sink and it made a huge difference:
https://www.amazon.com/gp/product/B00D1WYXR0
I wouldn't be shy about putting them in any closet, assuming it has power, and is normally enclosed. We're not on the wet side, so we don't have to go to quite the same length-- most of our storage is open, except in the kitchen.
I also tried rechargeable dessicant units, those help too, but the improvement wasn't as pronounced.
I had good success using small heater rods in enclosed spaces (cabinets, etc.), but I couldn't tell whether it worked when I put one under a mattress. It's a cheap+easy thing to try though. This is what I used, but I imagine there are similar products:
“These dehumidifier rods are made right here in the United States of America, and are perfect for using in a gun safe, on a boat or in a storage area to protect items from mildew and humidity.”
GoldenRod Original Dehumidifier, 12-Inch https://www.amazon.com/dp/B00D1WYXR0/ref=cm_sw_r_cp_api_G7fOAb05VNQ9V
Rust is caused by humidity in the air, and temperature changes that condense the water from the air upon the tools. Chests help with this as it moderates the temp swings, and also controls the humid air flow.
You can also get some bulk desiccant on Amazon (used for drying flowers) to put in cloth sacks that will absorb the humidity in the chest; or a "Goldenrod" device (used for gun safes) that you plug in. https://www.amazon.com/GoldenRod-Original-Dehumidifier-12-Inch-725721/dp/B00D1WYXR0/ref=sr_1_5?crid=2ABEHO17IIU2N&dchild=1&keywords=goldenrod+dehumidifier+for+gun+safe&qid=1628017385&sprefix=goldenrod%2Caps%2C211&sr=8-5
I don't have those extreme humid conditions, so I just have a bottle of Camellia oil and wipe them down after use. https://www.amazon.com/KUROBARA-Tsubaki-Japanese-Maintenance-Camellia/dp/B001TPFVHC/ref=asc_df_B001TPFVHC/?tag=hyprod-20&linkCode=df0&hvadid=198077680239&hvpos=&hvnetw=g&hvrand=17339704598038884629&hvpone=&hvptwo=&h...
A dehumidifier rod that lays in your safe, connected to power, and keeps humidity down.
GoldenRod Original Dehumidifier,... https://www.amazon.com/dp/B00D1WYXR0?ref=ppx_pop_mob_ap_share
We use these in some enclosed spaces, it's basically the same as putting a light bulb in there but it's more efficient: