alumnae badge
 

Is there a such thing, and do we get it from our chapter after we become alumnae

I'm pretty sure not.

Are you sure you're not thinking of the fact that alumnae can buy badges (used ones) from Nationals at lower prices?

No, there's not an alumnae badge. I believe there is a dangle that you can buy that specifically represents alumnae status. Also as an FYI any Phi Mu (collegiate or alum) can purchase badges that have been returned to EO for lower prices :)