I've done the unspeakable....,
As mentioned, I've had one or two spots that would not move on the extreme edge. I took two cotton swabs and covered the tips with new untouched Zeiss micro-fiber cloth. I blew the sensor (filter) as normal. I very gently with almost no pressure ran the cloth covered swaps over the extreme edges. rotated the tools for each edge. I did not go into the middle as there was no dust, and blew it off one more time. It is 100% clean! All spots including the one that would not move are gone. It was not oil.
I think we get a little crazy over this stuff. The industry would love us to spend $50 on sensor cleaners and special swabs.
I am not recommending my procedure, just sharing.
I did something similar but without the lint free cloth which resulted in leaving more fiber strands from the cotton swabs. Not so smart on my part.
I ordered a kit about four years ago and I've been cleaning my camera sensors ever since.
Re: Oil spots on sensor. Eventually, the excessive oil will be expended in due time and life will be much better. Like what the others have mentioned already, always point your camera down when changing lenses. It is what the owner's manual suggest and it makes sense.