I had that problem with my first reborn. Now, before I reborn them, right after I take them apart, I rub them down with straight acentone and some cotton balls. Don't let them sit in the sun with acetone on them, but if you're gunna do this, do it OUTSIDE. Stinks up the house and makes you see funny coloured lightballs. Wear gloves, and use a soft bristle toothbrush. (Though sometimes, the acetone will eat the brush bristles and/or the brush.)
Gently rub the acetone across the vinyl, wiping it away right after each stroke with a damp cloth. It takes a few strokes to get the rosy off the cheeks and feet, and about week and a half's worth of new skin growth to get the hair off. (So wear the gloves.)
When you're done with each part (ie: head, right hand, left hand, right leg, left leg, tummy, tushy) immediatly go rinse them in hot soapy water to get the acetone off. (I use baby shampoo so that the doll comes out smelling more like a baby and less like Home Depot.) DON'T LET IT SIT.
The factory sometimes leaves with bit of 'waxy' residue on the doll vinyl. That is where your paint won't stick. The acetone strips it and the factory paint right off.
Also, if you use oil paints, DON'T let them dry directly in the sun.
And, to get purple paint out of a doll part, take you half and half acetone and water, pour the limb about a third of the way up, cover with your gloved hand and shake like you mean it. Might need a toothbrush and some elbow grease, but it will come out. Believe me, I've done it more than once.
But, alas, once the vinyl has been stained, that stain is there. It won't come out. At least, I haven't gotten it out.
Thank you to LADY MAGE for this answer. :)