Paragraph 1 of your question: A part written for tenor sax is supposed to be and assumed to be transposed correctly. If the tenor sax part is in the key of A major that means the song itself is in the key of G major. That means that the written part will sound not only a whole step lower but also an octave lower because tenor sax sounds an octave and a whole step lower than written.
Paragraph 2: If you give a tenor sax player a chart in concert C (which is NOT a tenor sax part BTW) he has to transpose up a whole step and an octave to the key of D while playing in order to play in the correct key and register. The octave transposition is not as important as playing in the correct key. If the chart has chord changes and he is improvising he will also have to transpose the chord changes up a whole step in order to play in the correct key, for example, he will play a Dm chord as if it were an Em chord, etc. If he doesn’t transpose then EVERYTHING he plays will be messed up, he will be playing in Bb and everyone else will be in C.
I don’t know where you get A major from unless you are thinking of an Eb alto sax which transposes to A major for a song in C major.