Shark girl (nice name BTW), you may already know this (if so forgive me for repeating it) but this is a standard PADI practice. All instructors have the obligation to sign off on those training dives you did and pass on your details to another instructor and/or dive op. It's called a referral. PADI credits the referring party for doing so to facilitate the process. The instructor who does your last open water dive on a course is the certifying instructor.If time is an issue, it's worth looking in to taking the course as sets of adventure dives. I was able to do my AOW spread over a year; two dives on one evening in the local lake (PPB and nav), then three more the next dive season at the Poor Knights. They were done with two different companies- my local dive company were quite happy to pass my details on to the company up North with whom I finished the course.
This is why you will often see me post that if a diver is not happy on a course or has whatever issue, they should take their business elsewhere.