I would be inclined to learn or relearn old tunes the way I have done for more than 50 years.... I used fake books... They offered the chord progression, lyrics and a melody..

Use the melody line as a reference, the chords as the foundation allowing chord changes and bass lines..
Improvise within the chord progression in realtime..

If I have to learn a tune via an audio recording I try to work out the bass progression first, then chord structure..

Melody is not as important in your situation (other then reference), you have a sax player and possibly the guitar plays lead too..