It can sound complicated but it’s really simple:
the Bb sax is called that because when you play a C on it it sounds Bb on the piano (or any concert instrument)
the Eb sax is called that because when you play a C on it it sounds Eb on the piano.
That’s lucky you have a C melody, they have been out of production for many years.
They were invented precisely for the reason you brought up, all about transposing etc.
It was the sax that was not a transposing instrument, it was in concert pitch just like the piano.
