The best piece of advice I've ever gotten is: Don't stabilize anything that's wrong. My coach Andrew Murphy preaches this. I've watched lessons where super talented riders were told that they're riding too well, i.e. helping their horses out too much. "Don't stabilize anything that's wrong" aims at the same thing -- if your connection is not correctly reaching over the back into the bit, you can surely wrangle the horse round and "on the bit-looking" by more or less polite pulling on the bit. But in essence, you're stabilizing a fundamentally flawed contact. If your shoulder-in pops into a shoulder-out the second you let go of the inside rein, you're stabilizing something that's not correct, i.e. a horse not properly bend around your inside leg and on your outside aids.
Continually doing this doesn't work for the same reason draw reins (or sharper bits etc) don't work -- it doesn't teach the horse anything. It coerces the horse, but it doesn't create the desire to go there, let alone the correct alignment. It's like lifting your dog, who doesn't want to get into the car, into the back seat. It doesn't teach the dog to jump in the next time. It's just a quick fix that will need to be repeated time after time.
The second best piece of advice answers the "Then what do I do??" question. You wrangle the horse with your body, not your hands. That's action with your seat bone(s), with your correct positioning of your inside and outside leg, with lateral work on smallish circles... that's where your timing needs to be good, that's where your riding starts. And anytime I think a horse just doesn't get it and needs a little more hand involvement, it bites me in the bum in the long run. Sure, that ride I can convince myself that I've made some progress, but he likely won't come out tomorrow asking me if he can give me more. When it finally clicks though, and they start working correctly, they develop an enthusiasm that's infectious.
I am working with a number of older horses, some returning from injury, some coming from other disciplines, and just had one who finally used his back in the canter for the first time, surprising no one more than me. This horse went from a "oh dear, it's only been 10 minutes??" ride to a "shoot, we're done already?" And that's largely due to getting my butt kicked a little bit in the last clinic.
Just so it sinks in: Don't stabilize anything that's wrong. And that is, btw., also true for life in general: shitty jobs, unhealthy relationships... trying to fit a square peg into a round hole. (You're welcome. ;))