I have a leggy dracaena (I think it is a dracaena deremensis) with
about 7 stalks. The stalks are from 12 to 18 inches. The plant is
about 6 years old and the soil is worn out. What is the procedure for
starting new plants?

I have six dracaena out of a single one. I do not know the name, but it
is the one with dark green leaves and grassy trunk. The original plant
is still in its pot, with two of its daughters. when they become too
tall (6ft) I chop them half way and plant the top in the same or
different pot right away, all the way so that it touches the bottom of
the pot. The stump grows new shoots near the top. Amazing that such
large, leafed cuttings root so promptly.

I usually do this in May when I put them out in the open shade of my
front yard. To replenish the soil, every May I give them a couple of
inches of compost, as top dressing, and some earthworms to mix it all
in. This way the soil is never spent or compacted. If there are no new
plants in the pot, I may also give them one tbsp of urea, wood ash
and/or or 10-10-10.