To Bailey: These are definitely spots, and when I spot treat them with Tide HE Original, they come out. When I wash the clothes again, I get spots.
To AJ: We harvest all the rainwater that comes off our roof and store it in a 30,000-gallon tank. We then pump it through 25 micron, 5 micron, and carbon filters. Then it passes by a UV light and into the house. Our whole potable water system is rainwater. Water resources are precious here.
After I posted the question last night, I washed sheets with Tide HE Original but without OxiClean Versatile Stain Remover Free. They came out with spots on them, but not as many as previous loads. I purchased some Tide Washing Machine Cleaner and ran one package through the "normal" cycle with hot water. I haven't washed anything since I did that.
Since you say it probably isn't oil, maybe the OxiClean is cleaning residue left from the previous several years by hard water ... and perhaps the Tide Washing Machine Cleaner will remove the rest. However, the spots are removed by spot cleaning like an oil, rather than a mineral.
Our washer is Kenmore Model 110.45872400 purchased around 2005.