Maybe the following links will be helpful, though most of the information is about removing it from lawns:

Moss in the Garden

Bring on the Baking Soda

Moss in the Lawn

Lawn Moss Control [is] More About the Soil than the Moss

From WSU MasterGardener

Moss is taking over my garden and lawn. How can I remove it, and how can I keep it from coming back?

Moss is a symptom of several underlying problems. Too much shade, high soil acidity, low fertility, compacted soil, and poor drainage are all contributing factors to moss growth. Rake to remove moss in garden areas, then add compost or other organic matter to reduce soil compaction and improve fertility.

In lawn areas, moss can be raked and removed, or an iron compound, ferrous sulfate, can be applied to kill the moss. Whenever you use chemicals be sure to read and follow all label directions.

Once the moss is removed, reseed the bare spots and put your lawn on a regular feeding schedule. WSU recommends 4 pounds of nitrogen per 1,000 square feet of lawn per year, divided into four applications: April, June, September and December. If you can increase the drainage and decrease the amount of shade your lawn gets, you'll have fewer problems with moss in the future.

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Good luck! I imagine you may have more moss problems than many of the rest of us because of your climate and soil.

Thank you, Anney! These links are very helpful!
I think it's time to get a soil test and make specific improvements.

melgrow

The recommendations for using baking soda look like something I may try. We have moss on the northern side of the house where there's a pretty good overhang, so it's always shaded. I say "may try" because it isn't hurting anything. It's bare clay without the moss! So maybe it helps stabilize the soil.

Anyway, good luck again.