I don't think you should grease anything on this model, except from motor bearings. I would stick with oil if possible, you will get a smoother machine and it's easier to keep it clean in the long run. The reason for using oil is either because it's cumbersome to reach for regular applications (lids, screws, bottoms, etc, that needs to come off), or close to electronic/electric parts you can't afford the chance of a spill.
If it's more practical to grease gears underneath or something like that, I like Finish Line Ceramic Grease, it can be found in bike stores, and there's similar stuff available in various brands, Triflow too. I like the slighly opaque - transparent greases with teflon in small tubes or tubs, not the gritty lithium stuff, spray can stuff (usually white) or heavier sticky greases. The current Singer grease is a synthetic transparent type, but there might be more than one type.
For the stitch width knob I'm not sure. Is it a simple turn knob in direct contact with the parts controlling the swing arm? I'm thinking it might not be complicated like the buttonholer and stitch selection knobs in the 800 series. I can feel there are other parts sliding and turning against the setting knob on my 900 Nova. Is it possible to observe, check and poke the parts involved in the back?