Thanks to all for the many suggestions. This is a 2.375" diameter, 3"
long piece of Delrin. Yes, it is a bit slippery, and I suspect that the
rotation is causing it to slip a bit.

Yes, I have faced both ends, and when I do so, the workpiece length is
accurate enough that I can say that opposite sides of the cylinder are
the same length within a few thousandths of an inch.

The 3 jaw chuck seems to be accurate enough that I can't blame it; using
a micrometer, its runout is at most a few thousandths of an inch--not
enough to explain the problems that I am having--especially because I
can turn short pieces (1/2" long) and get acceptably accurate
centering--good enough that my cheap micrometer is less accurate than
the centering discrepancies.

Sherline does make a steady rest--but it only goes to 1.75" diameter, so
that's not an option.

Would it make more sense to mount the drill chuck in the headstock, so
that what is rotating is the drill, not the workpiece? Of course, that
means that I need to find a way to mount the chuck in the tailstock
instead, and I don't know if a Sherline lathe can do that.

Clayton E. Cramer