I think I've found an acceptable solution. I'm now using a table style AND
paragraph styles to get the result I need. The reason I don't simply use
paragraph styles is because the rows on our tables have alternating shading
(row stripes) and applying that manually to lots of large tables would take
a long time. The table style handles the shading nicely. Here's what I
did:

First, I created a table style that only defines the borders and shading I
need. In defining the table style, I made no changes to ANY font attributes
(I left them blank, so they don't even say Times New Roman or 12 points or
anything). Second, I defined the paragraph styles for the different parts
of the table (the header, regular text, the totals row, table notes).

To format a table, I first apply the table style, then I apply the paragraph
styles to the appropriate cells. This seems to be staying stable now and
the fonts are no longer spontaneously changing to random fonts from the list
of fonts.

Todd