grammaC,
The model # does not appear on the badge. Some of the later machines might have had a model # on them but I'm not sure the FW's did that.
Your's is a 221-1 rotary hook. Lift up the drawbridge on the end and watch the bobbin hook. Rather than oscillate back and forth it rotates round and round.
I get my Singer oil at JoAnns fabric stores. I got my last batch of Singer motor lube (grease) at JoAnns but they no longer carry it. Sew-Classic
http://www.shop.sew-classic.com/ does carry it for $3.95 per tube I think. I'll be getting my next tubes from her.
Yes you do need to lube the motor with the Singer lube. Depending on how much you use the machine you either lube it once or twice a year.
Sew-Classic also sells Tri-Flo oil and grease which I really like. It's a synthetic product that works very well. The grease is not for motors though, only gears and forks the like.
Joe