<xsl:template match="text()">
<xsl:value-of select="translate(., '&#xA;&#xD;', '  ')" />
</xsl:template>

But the above (as far as I can tell) applies to all my data. Do you have
any suggestions on how I can get it to only apply to data within the
<p>...</p> tags?

Try to restrict the tempalte match pattern:
 <xsl:template match="text()[ancestor::p]">
   <xsl:value-of select="translate(., '&#xA;&#xD;', '  ')" />
 </xsl:template>

A lazy way to avoid the template being triggered by text
in <pre> elemets would be
 <xsl:template match="text()[ancestor::p][not(ancestor::pre)]">