Most of your auto parts houses like O'Reilly/Auto Zone will carry white lithium grease. If it was my machine, I would open it up, take a little small paint brush and some solvent (auto parts house)or gas will work if you are careful and clean the gears real good, then put some white lithium grease on them and turn the hand wheel to make sure you get enough grease on each gear.
My mother-in-law used to run a Singer sewing center and that is what she did. It usually doesn't take more than 5 min. to do this and if the dealer is real good and not ripping people off, they will do this during a cleaning without charging extra. Cleaning can run anywhere from $75 to $150 on most of the older machines. If the machine is heavy, I would guess it probably has metal greases.