Chances are the more you try to convince your doctor to switch you to Oxycontin, the less likely it is that he'll do it. Pain doctors have to deal with so much drug seeking behavior & whatnot in their practice that I'd imagine anytime you bring up wanting to be switched to Oxycontin it will raise a huge "red flag" in his mind.
As somebody previously said, "climbing the morphine ladder" can take a long, long time. If you don't want to be on Morphine, next time he increases your dose you should just pretend that that high of a dose is having so many negative side effects that it's debilitating, AND that it STILL isn't working for your pain. That would probably be a much quicker way to get switched to a different pain medication.
Keep in mind, though, that if/when your doctor stops prescribing you Morphine, it doesn't mean he'll necessarily switch you to Oxycontin. There are many more pain medications that he could want you to try first. &, if managing your pain is your biggest concern, you may very well find that another pain medication works better for you than Oxycontin.