If you want to plug screw holes - sounds like a job for a plug cutter. It
will also let you cut cross-grain plugs that will blend in better. Should
be available from several sources. I bought a set of cheap JET cutters a
few months ago for around $30. Included four tapered and four straight
cutters (1/4" to 1/2" range if my memory is correct). Veritas and others
make more expensive cutters if you want a little more quality.

Otherwise the lathe suggestions are good. I find it hard to find walnut
dowel material locally and usually make my own. Just takes a few minutes
once I am set up.