One of the ideas present throughout the novel is that while Marianne believes that her brand of sensibility means that she has deeper and more profound feelings than others around her, in fact her reactions to situations where she is supposed to feel a certain emotion--joy, sorrow, regret, etc.--are all very much prescribed and calculated by the dictates of this type of sensibility.

In this case, bidding 'adieu' to her home means that she must wander around and forlornly say goodbye to every thing, lamenting their parting. It sounds phony because it is phony. By that I mean that Marianne already has this script running in her head of what she oughtto do and say in such a circumstance. She really does feel sad, no doubt, but her reaction is prescribed rather than genuine.

Check out this 1722 poem called 'The Force of Love'

Adieu ye Lawns! and every neighb'ring Grove,
Each conscious Witness of despairing Love;
Ye Rocks! whose Eccho's did my Sighs repeat;
Ye Streams, so oft increas'd by my Regret;
Adieu ye Flocks! your Master's fond Delight,
His Charge by Day, his tender Care by Night;
Some happier Swain shall lead your o'er the Green,
when lost Menalcas shall no more be seen!
Stung with the Rage of unremitting Pains;
In vain to Woods or Waves the Wretch complains,
In vain around these Plains I hopeless rove,
No Cure can heal the cruel Force of Love.