Hi Jeff,
I am living in Croatia in pretty cooler climate than you and because of warm summers I have hard time with moss growing to. As a reserve option ( If I don't succed with moss )I had planted a few Saxifrages ( Saxifraga X andresii) and Helxine Soleirolii.Basically Saxifraga is a mountain plant, and I have successfuly planted it in dry bed ( pH 7,5-8 )on full sun, where it needs to be regularly wattered. It has red blossoms in spring. It can easily be propagated by division in spring.You can try also with Helxine Soleirolii. Helxine is pretty fast spreader, and it covers big areas in a quite short time.Easiest way to grow Helxine is from seeds or by divisions too.Both plants will give you a needed mossy look.
Regards Myro

How about growing sedums? Some can look somewhat like moss and also are spreaders.

I also live in Zone 9a, but in Florida. We are building a house with a courtyard where I would like to create the serenity of a Japanese garden. All the Japanese gardens I have ever seen had plants that do not grow well in this zone: Japanese maple, small conifers, moss etc.

How do southern Japanese gardens look like? Where do I look for photos? I tried googling, but found nothing appropriate.