Less oil is better
Rag off the excess. I've read shotguns don't need a lot, and it's the oil that could cause a freeze up.
I know Vaseline (petroleum jelly) is recommended for bows as being scent free. I put some on my cheeks last week 11/20 (to prevent wind burn) and got a deer from a tree stand. I haven't heard about it used on guns, but it's worth a Google.
The old hunting books says scent control is over rated. I do my best anyway. Plan your hunt into the wind (easier said than done) and try to elevate your smell with a tree stand.