I can see 'happy/worried' or 'happy/anxious' but I think you are being a bit precious with the 'happy/evastated' bit.
Is it possible that your DS and maybe your DD too are feeling a bit suffocated and unsupported, respectively, as they move towards independence?
What sort of age gap is there between DS, DD and the younger siblings? Why are they so miserable about this trip? Have you and DH not made attempts to point out the major plusses to a round the world trip, not least the fact that DD has the gumption to give it a go? Have you made arrangements to skype or send postcards home, or little souvenirs? Assuming they are quite young, have you made an effort to get the younger siblings' classes involved in the trip by skyping? I recall one time when the brother of a girl in DD2's class did a stint in Antarctica he would send regular updates, video footage, etc. to his sister's class, and it was great.
In short, I think you and DH need to get on board unequivocably wrt this trip.
Your DH in particular has lost the run of himself. He most definitely should get over himself and go to the airport.
(I have brought children to university destinations that are more then 12 hour drives from home, and waved them off to semesters abroad on different continents, and saw oldest DD off to her first job, in a city it takes about 13 hours to drive to, in a hired car. Yes it is hard knowing I won't see them except perhaps for holidays from age 18 on, but they leave with my love and the knowledge that I share their hopes.)