You’re on the right track — it’s not a good idea to mix chlorine bleach with winemaking because free chlorine molecules can, if the conditions are right, contribute to appreciable levels of the swampy or stinky “corked” aroma, the most ubiquitous being 2,4,6-trichloroanisole. I never use any chlorine-containing cleaning compounds in my winemaking and don’t recommend it to anyone. There are plenty of alternatives for cleaning, one of my favorites being sodium percarbonate. It also sanitizes (depletes microbial cell count) on clean surfaces as it contains 13% available oxygen and is a strong oxidizer.
You’re also on the right track by thinking it might be better to clean your fermenting tubs thoroughly and then dry them out to store them for next time. In my experience in the cellar, even closely-monitored “wet storage” solutions lose their potency with time. Forget to add more sodium metabisulfite powder to your storage solution, go on vacation for two weeks and you’ll probably come back to a microbial house party in your tub — hosted by those same “airborne beasties” you were trying to avoid in the first place. While many strains of bacteria are sensitive to even low levels of sulfites, most spoilage yeasts and other fungi are perfectly happy cohabitating with sulfur dioxide under 50 ppm, especially if the pH of the solution is above 3.0.
The best thing to do — relax and realize that, after all, you’re going to clean and sanitize your tubs before you ferment in them again, right? Take a cue from commercial wineries, which don’t worry about keeping crush-specific equipment 100% sanitized year-round. If it’s not going to be used for a few months or until next crush, remove any visible soil, scrub thoroughly (being gentle with stainless steel), rinse, dry and put the gear away in a safe and dry place. Many wineries even have to store equipment, like large pneumatic presses, outside and just clean them, let them air-dry and then just cover them with a tarp until next crush.
For your tubs, simply wash them well with your chlorine-free cleaning compound of choice (I’ve even used dish soap), dry well and hang up, uncovered, so moisture won’t accumulate in nooks and crannies and grow any microbial bad guys. If you want to exclude dust, insects and other pests, feel free to use old sheets or large pieces of cheesecloth to loosely cover your equipment. Don’t worry about excluding airborne microbes since you’re going to be cleaning and sanitizing before your crush next time — it’s more important that the tubs be clean, dry and in a place where they’ll stay out from underfoot until next year!