Eh, Chas, is it actually oil, or dye? If it’s actually oil, I’d try cleaning them with saddle soap. They sell it next to the Kiwi shoe polish at the grocery store. You get a sponge wet (not sopping), rub it on the saddle soap and get some of it on there, then rub it on the shoe. The idea is NOT to scrub the shoe with soapy lather, but rather to sort of massage the slightly wet saddle soap into the leather. This acts to remove sweat and skin oils from harness leather, I don’t know why it wouldn’t at least help your shoes.
Then after it dries, you can buff it with a dry cloth if you want, if there’s white stuff on the surface of the leather.
If oil still comes out of the shoes, try doing it again. There has to be a limit as to how much oil a piece of leather can ooze, especially after four years.
If it’s dye, this might help, too.
If you get leather deck shoes really sopping wet, they dry stiff.
This website http://www.stretcher.com/stories/00/000410c.cfm suggests using Goop. I’ve never tried it myself, but I have used saddle soap.
Were they actually Rockport brand deck shoes? I’m surprised, I never thought of them as cheap shoes.