This method is sometimes referred to as a "swamp cooler", and is well known and used in homebrewing circles. Honestly, if the brew shop employee told you it wouldn't work then they are either (a) trying to sell you a brewing fridge, or (b) not that educated on homebrewing.
Change out some ice packs in the water twice a day and you get get down to the low 60's for the fermenting temp. For a 5gal carboy, I think you'll want the water in the garbage bin to reach about the 4gal line. Be sure to stick a thermometer on the side of your carboy/bucket, at about the 4.5gal mark, so its above the water level, and below the wort level.
If you are worried about mold and mildew, you can add a tbsp of bleach to the water every day or two. That should keep the water fresh for a while. Plain water isn't really going to attract the really nasty bugs like acetobacter, so your vodka airlock will be more than sufficient to protect the beer.
Temp control is crucial to beer quality for about the first 3-4 days, and less important after that (for ales anyway). Make sure you are super vigilant about changing the ice packs (frozen 2-liter bottles work good too) for that time, and then after that, you can relax a little and maybe just change them out once a day or two. I wouldn't let the temps get above 72-74F, if I could help it.
If you don't want to swap out ice packs, drape a kitchen towel or a t-shirt over the top of the carboy so that about half of the fabric is in the water. Put a small fan to blow directly on the shirt. The shirt will wick water up and the evaporation that occurs will steadily hold down the temps too.