Answer: I understand your methodology is loosening the pins which results in softening up the plastic housing from excessive heat. Another way to remove the solder is to suck it out with a hot air tool offered by manufactures such as Kurtz Ersa, Metcal or others providing tools of this type. Size the tip to the style of connector being used and this will reduce your problems as the hot air tool will pull the solder directly out of the cup.
If that is not an option for you, manual solder suckers are definitely better than using braid solder, as less time is being used to reflow the solder in the cup and remove it resulting in less heat application and possible softening of the surrounding connector materials.
The other option, if you have the luxury, is to get the components with gold pins, without the gold cup, so they do not need to be processed at all as you have mentioned above.