If I dab a little solder from the rosin core solder wire onto my soldering tip, does any of that flux make it onto the surfaces to clean/wet or does it cook off pretty quickly?
Don't put solder directly on the tip of a soldering iron, unless you want to pre-tin the tip and clean/wipe it off.
Rosin is needed to solder and it will quickly burns off, making the tin useless.
Heat one of the pads with the tip, and apply a tiny amount of solder where the tip meets the pad.
Quickly, so that there is still rosin left on the pad.
Position the LED with tweezers, and tack that side of the LED.
When happy with it's position, solder the other side of the LED first. Then re-solder the first pad.
Solder quickly (<3seconds), and wait 10 seconds between (re-)soldering.