Certified GKH Gardening Expert
This is an infection, rather than going woody. Woody stems are a natural part of aging, and will occur, only, on older growth. This new growth that is effected by a disease. This, usually, happens with overwatering or soil that doesn't have a chance to dry out thoroughly down to about 2 or 3 inches between watering.
I would treat with a fungicide or change out the soil and treat (which will be more thorough).
Here are some articles that will help: