Is it okay to store chips and modules in a plastic parts organizer without anti-static bags?

One sachet per cabinet won't do much in a cabinet like that, because those drawers are not sealed at all. Air moves freely between compartments and the whole thing breathes with the room, so the silica in one drawer just ends up drying the whole cabinet a tiny bit and saturating fast. I live maybe a mile from the water and went thru the same rust problem with pin headers and resistor leads. What actually worked for me was a couple of the bigger 50 gram pouches sitting loose in the cabinet, plus recharging them in the oven every month or two. If a drawer smells musty or the sachets feel heavy and saturated, swap them out. You can also get indicator beads that change color when they're spent, which saves you guessing. The other trick that helps a lot is keeping the cabinet in the driest room you have rather than the garage or shed. A closet in an air-conditioned room beats any amount of silica sitting in a damp space. And whatever you do, don't repeat the cardboard box thing. Cardboard wicks moisture like crazy. On the static question, loose chips rattling around in a plastic drawer is genuinely not great. Plain polyethylene drawers are insulators and they can hold a charge, so a MOSFET or a CMOS chip sitting in there for months with its legs touching plastic is a small gamble every time. The cheap fix is conductive anti-static foam. Buy a sheet, cut strips to fit the drawer bottoms, and stick the leads of your sensitive parts into it. That's how parts shops store bare ICs and it keeps every pin shorted togehter at ground, which is exactly what you want. It costs a few dollars and lasts forever.

For the boards and modules, I'd bag them. Not because they'll die in the drawer tomorrow, but because a shielded bag also doubles as your moisture barrier, and it protects the pins from getting knocked about. Regular parts like resistors, ceramic caps, and connectors can stay loose, honestly nobody bags those. But anything with a FET input or a micro on it goes in a bag or in foam in my setup. Cheap insurance for parts that cost real money to replace.

Might be worth grabbing one of those little digital hygrometers and just leaving it in the cabinet for a week. Costs next to nothing and you'll acutally know if the silica's doing anything instead of guessing. I did that after moving my parts bin and was surprised how much the humidity swings day to day. Also don't panic over slightly tarnished leads on older transistors and such. A quick swipe with a fibreglass pen or even fine emery and they solder up fine. I've rescued loads of parts that looked way worse than a bit of rust on the legs.

The thing that bit me over the years is that static damage is often latent. A board survives the drawer but takes a partial hit, works fine for months, then dies randomly and you never connect it back to the storage. I run the component store at my job and we keep anything static sensitive in a dry cabinet set to 10% RH, mostly because moisture and ESD compound each other. At home I just use a lidded tote with a couple sachets and the shielded bags, no need to get fancy. Worth checking your drawer drawers too, some of those cabinets come with conductive liners and some are plain poly, they vary between batches.