There is a chance that you are using the soft center intended to be used in the headstock, in the tailstock, I'm assuming that you have a MT sleeve in the headstock and the centers are the same size MT. When you need to true up the soft center, insert it in the sleeve in the headstock and true it up with a cutting tool, on my lathes, I made match marks on the spindle nose, the sleeve, and the center to assure concentricity.
I am the source of the outlawed white lead, it is good for centers, and especially for press fits to avoid galling. I think McMaster Carr sells CMD center lube, it is an extreme pressure grease, and it is also good for steady rest lube.