Here's the manual for your 403A
http://www.singerco.com/support/inst...manuals/search/instruction-manuals/search
You will need it so you will know where to oil and grease. You're going to need some grease for the gears. I used some Singer lube that I had on hand, but someone here can tell you what other types you can use.
Yes, you will need the zig-zag cam to ZZ. Lucky me, I have three of these machines,(don't ask me how that happened
) and one of them came with a full set of cams.
Janey, the 403 I'm working on has two cords, one power and one for the foot pedal. My other machines are in storage, so I can't check them and I don't know if the 403's were also made with double cords. I have a Singer 301A that uses the double cord with one plug, but I've heard that some of that model were made to use two cords.