This seems pretty basic.
...If String1 has a value then Type tag should have value as "axbx" and if String1 is blank then Type tag should have value as "dydy"
Code:
<xsl:template match="String1[. != '']">
<Type>axbx</Type>
</xsl:template>
<xsl:template match="String1[. = '']">
<Type>dydy</Type>
</xsl:template>