Most of my work is flattened so I'm well-aquainted with corks and the smell of them burning. Corks aren't the only tool to flatten glass; other materials you can use are metal (which leaves chill marks), wood (can stick) or graphite (can also leave chill marks). After finding some of the commercially available corks lacking good handles and only available with rounded edges I began making my own. I found there were economies in buying giant cork slabs and making a dozen at a time so I could sell the overage to other glassblowers. Selling them generated some questions from buyers who hadn't used corks before.
Here's how I use them:
When they're brand new I burn them in before using them, lightly charring them in front of the glory hole or a fluffy torch so they have a thin layer of carbon that will touch the glass rather than the new cork. This layer of carbon will help them slide (like a block).
I use mine totally dry, while some people wet or moisten them first. I prefer to use them dry because I run no risk of scarring the glass with a dripping cork and the glass moves a bit more while you rub it since dry cork isn't as chilling as wet cork. I think ppl who use them wet feel like they won't burn up as fast, but I generally get 3-4 years out of a set (which seems like a long time), so I'm not concerned about longevity.
I generally flatten vessels while my assistant is capping so the bubble doesn't dent in. This lets me get the piece a bit extra flat without the pressure of the cap, but also keeps the denting in to a minimum. Right after a strong even heat I'll have him stop and cap and while the glass sags on the pipe, I rub the bubble between the paddles in a circular motion. Since the glass is sagging, I can only rub for about a second before so my assistant needs to flip the piece and I flatten again. We keep doing this--cap and flatten, flip, cap and flatten, etc. until it's as flat as I desire. If it's really tough to flatten a piece, get it hotter. If it's getting too wide or if I want it extra flat, I'll have my assistant uncap for a second--this will let the vessel flatten quickly under the corks without internal pressure pushing back. Uncapping gets things flatter quicker but the glass will usually dent, so either uncap only very briefly or give it a quick puff to re-inflate and re-flatten. I move quickly and firmly on the glass with the corks and keep them moving constantly, this prevents the cork from sticking and imparting texture to the glass. It probably helps them live a long life as well. The paddles always smoke and occasionally flame if I take too long. They don't seem to mind.
The corks eventually will begin to burn out, but in the center first. This will create a dished effect on your corks; when this dishing gets deep enough to become a problem, I'll sand them down on a belt sander to re-flatten them, then re-char them. If I find the corks feeling a little 'tacky' (usually after 2 years or so) I'll also sand them a bit remove a layer of carbon.
I'm generally using these to flattening vessels that end up between 15-26" tall x 10-16" wide x 4" deep, so medium to larger work. I've seen people use two newspapers (one in each hand) to flatten work as well. I like corks better because they're flatter, shield your hands better and there's little risk of them slipping out of your hand while you're flattening