Good job! Halfway around the ring is a start!
If you and your horse don’t mind the dropped stirrups, maybe try doing sets? Like, don’t cross them so you can pick them up. At the posting trot, go halfway around without stirrups, then pick them up for a lap, then drop them again for half the ring, and so on, both directions. Or give yourself some breathers so you don’t over-do it.
Also, try putting reins in outside hand and pulling up on the pommel with your inside hand when you sit the trot. See if that helps you sink into the tack. You can try it at the walk too – also try 2-point without stirrups at the walk. And 2-point with stirrups at the trot, until you can drop your stirrups for that too.
For some of the bouncy ones, I think to myself to go with their movement, rather than try to sit still, if that makes sense? And having them go forward, carrying themselves (not faster), makes them a tad less bouncy/more swing-y, especially when you can get on a rhythm.