I have an offer at another university. It is late June, just got it. They want me asap. I have asked to not go until after the fall semester as we have an accreditation site visit as well as a faculty member coming back for their first semester after cancer treatment. I don't want to stress out the department with a job search now and to go through the site visit without me. I literally have my current university's contract in my hand and have not returned it. I would like to ask them if I can accept for just the fall semester. That would give us time to get through our very important site visit, and the fall semester. It would also provide 5 months for them to fill my position. My fear is if I ask they could say no. If I don't ask, I'm not sure what if any penalties there would be. I want to do the right thing but not sure what to do. Suggestions?
But the decision appears not yours to make from my reading of the wording of the question.
If the current institution says no to a one semester contract, then the answer is given, as it would be kind of ridiculous to keep putting off the new position (assuming one wants to accept it) for however many years the old place keeps renewing the year-to-year contract.
But lets imagine that the OP is in a position where he/she is negotiating with both sides to stay one semester. The former department has requested that they 'need' the OP just a little bit longer. Accommodate the request or not?
There is no ethical obligation to stay (how would people ever change positions otherwise?). A junior faculty in a field where everyone knows everyone and a collaborative team player reputation is highly prized, might find it a smooth move to attempt to negotiate to stay one last semester at the desperate request of the former department. (If the new place says no, well then, figuring that out as soon as possible is more important to the old department than anything else).
But, even so, I get a sense from the question that the OP feels guilty for leaving, (which is normal). I also suspect that is it pretty easy to mistake anxiety-over-getting-out-of-project with being indispensable. ("It won't be done the way I would have done it" is not the same as being indispensible)