I had the same issue with my Flues.
I decided to leave and give it a good wiping after use. I have had no rust after 5 years of use. It is a shooter and gets in the field often.
I have used several cold blues and while I think it has its place I did not think this was it.
BTW, I too like Singer sewing machine oil.
My Flues is a 16 ga. and perhaps my favorite gun. Use it and enjoy.
Good stuff.
MIke