is confusing and makes no sense to do.
Anyways, I have this same problem. My fretboard always gets dirty. Everyone else I see who plays guitar doesn't seem to have this problem. Even changing the strings like you said doesn't really solve the problem. I end up having to scrape it off with the side of a pick when I change my strings.
I've also heard some negative things about lemon oil. Some say that it works good at first, but eventually it dries out and dehydrates the fretboard. I also heard you shouldn't use it on maple necks so you might want to think twice and do some research before you use it.
As far as applying any types of oils though, a lot of the time it is mainly for "hydrating" the neck to keep it from drying out. You don't want to overdo it. Only once or twice a year is needed.
Oh and I also heard that Old English is "fake" lemon oil
edit: As far as what I use. I have Dr. Ducks Ax Wax and it works great for keeping the fretboard hydrated. Wouldn't necessarily recommend it as a cleaner though because like I said, you don't want to "over-hydrate" the wood.