Av for small signal analysis with BJT for unbypassed emitter and ro in place
Your result has no problem, and your procedure is all right. To verify this, you can substitute the \$Z_{b}\$

$$
Z_{b} = \beta r_{e}+[\frac{(\beta+1)+R_{c}/r_{o}}{1+(R_{C}+R_{E})/r_{o}}]R_{E}
$$

into your result and into the textbook's result, and do some simplification, you'll find they give the same result. Cheers!