aforistall, an excellent reference for Featherweight information is
https://singer-featherweight.com/. There is an excellent tutorial on oiling and lubricating.
The Featherweight originally had a walking foot made for it - the Penguin. Do a search for it on the above website for "Penguin" to learn about it. It is highly collectable and expensive. That website sells an aftermarket walking foot made to fit the feed dogs of the Featherweight, and the price is reasonable. Just do a search on the website for "walking foot".
Some people have had good results with generic short shank walking feet, which are readily available. However, the generic walking feet are not exactly aligned to fit the unusual narrow design of the Featherweight's feed dogs.