Mujitsu and Tairaku's Shakuhachi BBQ

Get a chromatic tuner and use it. It will tell you where the pitches are. Any exercise or piece or phrase can be used.

You are talking about jazz and western pitches, correct? So we assume that you are hitting the main ro, tsu, re, chi, ri pitches accurately (except that since you have a vintage flute your chi may be a bit sharp).

Practice intervals. James Schlefer's book and CD has a lot of great exercises on that subject. Fourths, fifths, sevenths.

Another thing you can do is just pick a certain interval and work on it. Most important one would probably be Ro to Tsu meri. There's a lot of room for error there. If you're playing jazz, define tsu meri as Eb and just go back and forth between them until you nail it every time. Then pick another interval like re to chi meri and do it. Choose all the difficult intervals and hammer away at them until you own the intervals and can't make a mistake.

If you want it to sound Japanese even if you're playing jazz remember that a general rule is the strong notes (ro, tsu, re, chi, ri) are  played straight and then vibrato is introduced during the course of the sustain of the note. Meri notes have vibrato from the inception. This also helps keep them (sounding) in tune, although I suppose a note with vibrato is by definition not in tune all the time, LOL.

INTERVALS! The better you control them the more in tune you will play.