I have a few irons in the fire as far as jobs go. I have one offer so far and am waiting to hear on a few others. I am working hard to finish up my PhD, but might not have it completely finished by the time the Fall semester starts. That being said, I am optimistic about finishing and have heard encouraging things to that extent from my adviser and other faculty members. How should I broach this subject with schools after I get the job offer? I want to be honest and up front that there is a non-zero chance that I will not finish before the position starts, but I don't want people to second-guess me when I might finish on time. And all of these positions are adjunct/lecturer positions if that matters. I also didn't get a masters on the way to my (eventual) PhD. So when and how should I bring up this subject at least for the school I got the offer for?
If this were a full-time tenure-track position, the answer would likely be different, because in many fields (particularly STEM fields) you'd probably be able to request a "gap year" that you could use to finish up your work and do a short postdoc stint somewhere before beginning your full-time position.
However, as a lecturer or instructor, you don't have nearly enough leverage. You may ask if it's possible to start the position if, for instance, the thesis has been submitted but not defended by your planned start date. But I would get in touch with them as quickly as possible and ask what would be the latest you could get your degree before starting. Then, if you feel you can't make that deadline, try to negotiate (but be aware that it might not work!).