Some thoughts from a flutist with arthritis in the hands (now in remission, hopefully to stay that way):
You don’t grip the flute. Grip implies muscle tension and strength. You balance the flute.
No part of your body should hurt. No set of muscles should be particularly tense. Speed and accuracy of fingering is impossible if the fingers, wrists, or even elbows are tense. Efficient breathing is impossible if the shoulders, arms, or chest are tense.
Different flutists find different grips that work for them, and there is no “one right way”.
That said, the approach that works for me:
There are 3 points of contact to balance the flute: the chin, the base of the left index finger, and the right thumb.
The chin (or the lower lip, depends upon how you look at it) is a fulcrum. The flute rests gently against it but isn’t held up by it.
The base of the left index finger rests on the side of the flute away from you, and presses gently in. It doesn’t go underneath the flute, and it doesn’t “hold the flute up.” The left thumb can either hang free or be held gently to the flute but it doesn’t go underneath the fute and it doesn’t hold it up.
The end of the right thumb rests on the side of the flute closest to you, about midway between the first and second finger right hand. Again, it doesn’t go underneath the flute, but rather against the side of it, pushing gently out.
Try to keep your wrists straight. This will bring the elbows up and away from the body, and that’s ok: that helps with good breathing and good posture in general.
The head should be held normally, and the neck should be relaxed.
That’s the approach that works for me, and for many other flutists. If you ever hear a flutist talking about the “Rockstro grip” or the “James grip,” this is it. (Note: I’m not that James.)
Best wishes, and keep working with it! You’ll find a way that works for you.
–James