How to remove an oily stain from raw wood before painting

It sounds like the dampness is still a problem in that particular area because it is a corner adjoining the external wall and the store area. Is this part of the house in shadow for a lot of the day?

1. Check the external ground area surrounding the outside store. If the ground is clay or compacted it might not be helping in that it is storing water. Perhaps better drainage there.

2. Some form of heating in the store area to keep it at a similar temperature to the kitchen e.g. condensation happens in that part of the kitchen because of the cold store area on the other side of that particular wall.

3. Perhaps a better seal on the door and windows in the store to keep it warmer in general.

4. Put a faux wall in the back of the store to buffer the cold air away from the wall that is adjoining the kitchen.

5. A robust kitchen extractor to remove as much steam as possible from cooking.

If the prior owner used caulk, and they were really dumb and didn't use paintable latex caulk, then its possible that this would be the result.

Regardless, obviously the paint isn't taking properly here - so yes, the answer here is get a quart of primer go over the edges with a brush, let it dry, and redo the corners.