A key thing to remember about an offer is that it isn't an agreement. It is a proposal for a contract between two parties of which at any point either party can accept or decline. The terms may have been discussed and negotiated but agreement only occurs through ratification e.g. by signing or, arguably, by handshake.

There is a reason most people feel bad when they have not been offered a job, an assignment, a counter offer etc. That reason is that they are all forms of rejection. The rejection is not necessarily of the individual though but maybe of the situation regarding the individual.

Act professionally and all will be OK. If you are honest and upfront with all parties then any reasonably party will accept any outcome. Due to their own current pressures they might be off hand or even rude at the instant of rejection, however, in the longer term they will be more understanding. If they are not reasonable then you should not concern yourself.