I was able to get some ESRI support time with Jon B. from ESRI (thanks Jon!). I shared my screen with him and we walked through the processes to make sure they worked.
The solution was to go to the "Geoprocessing" menu> "Environments..." > "Processing Extent".
Under Processing Extent choose a layer that you know covers the full extent of the waters you are processing and click OK. This should ensure that the processing extent is large enough to do what you want.
To check this, I purposefully choose a Processing Extent to match that of my Batch Points and ran the subwatershed delineation. The result was like the original, the limits of the subwatersheds were constrained by the extent of the Batch Points. When I changed the extent to the rawdem layer and reran the subwatershed delineation, the results were correct.
So the solution was simple. I'm still not sure why the processing extent was set to that of the batch point layer when I encountered the problem.
