Don't use too much water, it may even prevent you from melting the solder itself if it's too close to your work area .... plus, cooling a solder joint too fast can result in a "cold solder" joint or a crystallized join, which is quite fragile and easy to break off. The use of an aluminum heat sink to absorb the excess heat is the better idea, but so is keeping parts in place with galvanized steel wire or even aluminum wire... the solder wont stick very well to these metals, and is a fairly easy cleanup with a steel wire wheel or a grinding wheel with a rotary tool.
I would put parts in place and wrap them down with aluminum wire before and between soldering the major components in place ... work your way down from the largest pieces to the smallest, any time you can.
Keep flux only in the areas you intend to solder ... a small watercolor paintbrush works well as a detail applicator for the flux. Uncleaned copper/brass won't have nearly as much solder adhered to it. Once everything is all in place, a steel wire brush in a rotary tool will remove excess flux and shine up the copper and brass ... finish up with a buffing wheel or cone and red/white rogue.
Alternatively, you can leave it looking more of a mess, and even add more to give it the aged, used and beaten up look, which is very popular for steampunk. Either way, lead based solder is poisonous, and is considered a cause of cancer, so you really should seal your creations as much as possible in some kind of safe encasement, such as a clear epoxy or polyresin... silmar or alumilite work too. This would definitely also preserve your weathered appearance, if that's the look you choose. (Even if you use the modern silver based or "plumbing safe" solder, it still contains a large amount of tin, and bronze contains appreciable amounts of lead, if you use any. Better to be safe, than sorry.)
Steam can be a lot more dangerous than many people realize ... it can burn you before you feel it. If you use the wet or damp towel trick, you'll also want to make sure you keep it damp, because if it dries out too much it will catch fire. Fire + wood shop = very sad news. All round, the aluminum heat sink or wire method is a good deal safer to work with.
Keeping a damp sponge nearby helps you with keeping the tip of your soldering iron clean. You will want to periodically wipe the iron tip on the sponge to remove any blackened buildup, as the carbonized/oxidized material insulates the tip from making contact or transferring heat to your work. If it doesn't come off, try a steel wire brush ... if it's still being stubborn, allow the iron to cool and use a file to remove the buildup.
Once the tip is clean again, plug the iron back in and properly tin the tip with flux and solder. As the tip heats up, either paint the tip area you wish to use with flux or dip it into your flux container, and then melt solder directly on the tip as it reaches operating temperature to coat the copper tip with solder. This layer of solder provides several functions ...
1. it assists with heat transfer, by creating a liquid bridge between the soldering tip and the work area
2. it protects the copper clad tip from oxidation, which is greatly increased by heat ... the tin keeps the lead from oxidizing as well.
3. it aids in the flowing of solder into a joint, by removing areas of the soldering tip that solder won't adhere to and gain heat from.
4. it protects the tip from corrosion when the soldering iron is not in use. (Make sure to glob some extra solder onto the tip for a thick layer before you unplug it and let it cool off, and then put it away properly after it's cool.)
(These are all tips that are known and commonly practiced in the electronics repair trade, but it also works well for plumbing style soldering irons and other related tools ... stained glass irons included.)