Bismihi ta'ala 

It is indeed a dilemma many of us face. Some Muslims have resorted to showing too much leniency in this area, and saying we need to get with times and shake hands with those of the opposite gender who are non-mahram. One of the misunderstandings is that it is a matter of cleanliness, where the female is seen as "unclean", and hence the male should not shake her hand. Another is that it is a matter of sexualising the shaking of hands. 

As Muslims we know our boundaries when it comes to non-mahrams, and it is natural for us as well. We just need to have a sense of pride and respect our Islamic culture and religious laws. We also know that if we are in such a predicament and wont be able to explain or might result in distress, or have time to decline, then it is permissible for us to shake hands. 

 I have a video tutorial on this issue which might be useful, where I give alternative ideas when dealing with this. 

With prayers for your success.