What kind of soldering iron is it? You can try to use a pair of pliers to remove the tip while the iron is still hot. If its a cheap iron, I would say just go get a new one, preferably a quality one with variable temp. (more than low and high)

The flux will float to the top of the solder. No flux will remain under the solder, as long as the solder is flowing and sticking to the components. I attached a picture to help. As long as the solder looks like the first two sections, then there is a connection. Too much solder, while not ideal, still makes a connection. What you are describing is De-wetting, that is when the solder does not stick to one or both components. In that case flux will remain in between the solder and component, but it is not caused by the flux. The flux is there simply because its available space to be in. If you have de-wetting, it will be obvious, and the components will pull apart easily. Like I said before, the only time that you would have that problem is with old components that are not clean and there is too much corrosion for the flux to remove. The parts would look discolored and obviously dirty/corroded before you even try to solder them. Unless you are trying to use battery connectors you found from 20 years ago or sitting in salt water for months, you have nothing to worry about with flux getting under the solder. It simply can not happen, If the solder is not sticking properly, then you have other issues, like oil, corrosion, or some coating preventing the solder from sticking.