There could be several reasons your playing doesn't sound right:
the music you downloaded might have mistakes.
the music you have might not be transposed for tenor sax. The notes need to be transposed a whole tone higher. The key signature will have two more sharps (or two less flats).
You might not be playing the notes correctly.
there might be something ...
It has been known that stuff on the 'net isn't always accurate!
There's also the possibility you haven't taken the key signature into account.
There's also the possibility that it's your own mistakes, touching the wrong keys on the sax.
Answering the header directly - you look at the key signature - the sharps or flats at the beginning of each line, and ...
How old is the instrument?
A440 was only adopted in 1939 and although older instruments can play in tune to an ensemble tuned to A440, or in actual practice in most high schools, Bb, 233 hz, the fact is that they can sound "pitchy".
Correcting this is a bit more complicated than just tuning the instrument with the lead pipe. You have to have ears ...