Academic changing universities: How to negotiate start date to minimise problems with leaving previous university with little notice?
Also look at Possible to leave a contract academic position shortly after signing new contract? [closed]

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)