I did TRY calling it, but must not have done it correctly. Were would I call
it from, Tom? I would like it to apply to all output content.

Try a template for text:
 <xsl:template match="text()">
   <xsl:call-template name="escape-apos">
      <xsl:with-param name="string" select="."/>
   </xsl:call-template>
 </xsl:template>

Change your template to
 <xsl:template name="escape-apos" match="text()">
   <xsl:param name="string" select="."/>
   ...