Hard to tell what that white residue is from the description. It could just be old soaps or at worse efflorescence. Try using a mixture of 2/3 water, 1/3 bleach and some TSP. mix a small batch and test a small area. Be sure to rinse it well and see it that gets rid of it. If it doesn't do the trick, you may have to etch the exposed concrete with mureatic acid and rinse well before repainting.
Assuming you finally get it cleaned up, let it dry well. Use fans, dehumidifier or ventilation to speed the process. Then I'd really recommend a good bonding primer and overcoat with a good flooring enamel. You can use regular concrete paint if it is not a high traffic area, but I like using a strong bonding coat first.
There are (at least) two different types of stains for concrete. One is a surface coating, the other is acid etching.
The former can work, but isn't as durable/lasting. I believe most of the products you get at the big-box stores are more of a top-coat type of stain.
Acid etching is just that, it's a stain containing an acid that actually etches the concrete. This is the typical type of stain you see in home magazines and such. The etching provides a deeper stain that is more durable and not prone to flaking.
I've done the latter, and applying it is super-easy. It's the prep work that can be a chore--namely if you have adhesive on the concrete already. In our case, we has tile mastic, which we had to dissolve using soybean based stripper. It worked great but was incredibly messy.
If you don't have glue to deal with, however, you should be good to go. Order up the stain (lots of places online to get it). You then dilute it per instructions and spray it on the concrete. I used a garden sprayer for this. You then let it sit for a period of time, and then you apply a neutralizer to stop the reaction.
After that, you apply water and mop/squeegee/wet-vac it up.
Then, you apply a sealer of some sorts. Common options are wax-based, polyurethane, and epoxy. I went with poly as it was relatively easy to apply (roll it on). That said, I don't think it was ever truly waterproof. For your needs, you may want to go the epoxy route.
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The smell from any paint or varnish is a result of outgassing. Leave the doors open and that should disappear in a week or less. If it doesn't, or the paint remains "tacky" then strip and redo with fresh paint.
Unless you are storing foodstuffs unbagged, there's no direct contact between food and the drawers, so it's a nonissue. If you are storing loose items (onions? ) I still strongly doubt any significant material transfer can take place. If worried, peel and/or wash before use.