For a RHBH,
I've heard lots of different tips. Some say to remember to keep your thumb down when releasing.
Of the two, do you think it is more having to deal with your wrist flick or your body/arm motion?
One thing and one thing only for throwing a RHBH anny...tilt the wing of the disc up! Coming out of your hand it will look more like \ as opposed to a hyzer release like / just less extreme than those two angles.
I've played around with annies a lot and it doesn't take any weird body movement to accomplish the flight. Just tilt the disc, the rest of your throw can be exactly the same as it normally is. Same x step, release timing, power etc.
Regarding disc selection, you can make any disc hold an anny line. I used to drive holes 3 & 4 in Canton, Ohio (Both right turns) RHBH with a max weight Firebird.
Generally the discs I use for anny drives (Assuming it's out of Wizard range) are more stable. Worn Teebird or heavier star Leopards. Super flippy discs feel less controlable for me.
These days I drive hole # in Canton, OH with a Buzz, and it works well.
I find throwing an anny with a disc that has a bit of stability can help me predict the landing better. As long as I don't overpower the throw, I can make the disc hold the anny line, then gradually straighten out as it comes to the ground. Still fading right when it lands, but the disc is about flat when it lands so it doesn't turn into a roller. That's the thing I don't like about real flippy discs, trying to anny with them will turn into a roller often.
I don't find much use for turnovers but when I throw them that's when I like a flippier disc. I have some super worn Eagles, Teebirds etc that can be thrown 325 feet with lots of hyzer and they turnover easily. Good for tight, right turning drives but not much else in my experience.