I wish I knew what the answer to this is, my son gets it a lot with his clarinet. If you find something, please let me know.
As you well know I'm quite the neophyte when it comes to playing a sax, and I never did get a good handle on the clarinet. Just the same I have "my answer" to your question, as least with respect to the sax.
What I've discovered is that I was simply using too much pressure on the base of the reed with my lip. Way too much! I think there are basically two ways to play a reed.
1. Using brute force
2. Using delicate finesse
The Brute Force Embouchure
I think there's a tendency to use the brute force, because there tends to be more control that way (i.e. more confidence, more consistency) and in that sense it seems to be both, the "better" way to play, as well as the "safest" way to play (i.e. less chance of getting unwanted squeaks, squawks, and reed "drop out", etc)
The brute force method is where the lower lip is placed at the base of the reed. Where the reed meets the table of the mouthpiece. Then a lot of pressure is used to control the reed from there. It takes quite a bit of pressure to control the reed from here, but the control is quite stable.
However, I think in the long run this this stability isn't worth the effort. It may be true for raw beginners (i.e. they can actually play this way and have quite a bit of control in the early going), whereas they my have difficulty using the finesse method (addressed below) and still retain control. So they have a tendency to fall into the "brute force embouchure position" just out of pure habit. It works, and it's what they have learned works, so they stick with it, far too long.
The Finesse Embouchure
The "Finesse Embouchure" (my own terms by the way), is to release the pressure of the lower lip, and move the lower lip upward (toward the tip of the mouthpiece - or more "onto" the reed). Onto where the reed begins to leave the table. It requires less pressure from the lower lip to control the reed from here. But it also requires more "finesse". The pressure and position of the lower lip has a much greater effect on the reed because it is now pressing on the reed where the reed is more vulnerable to that pressure.
Thus it's more difficult to control the reed. This actually gives you more control over the reed in the long haul, but it's more difficult to control the reed (especially for a beginner) because slight variations in lip pressure will have a major impact on how the reed plays.
So that's what I've found for me personally. I don't know if this makes any sense to a professional saxophonist, but it seems to work for me.
It's just the precise control of the lower lip pressure and placement at the base of the reed.
We're talking about very small variations in the placement of the lower lip. Moving the lip further toward the base of the reed requires more lip pressure but results in what seems to be more stable control. Moving the lip ever-so-lightly outward toward the tip of the reed requires less lip pressure, but also seems to 'lose control' as well. But actually learning how to control the reed like this using less lip pressure is the key. It's probably the key to better playing overall. But at first it will feel like "losing control".
So I just consciously try to work my lip as far back (toward the tip) of the reed as I can get it and still retain control. The further back you move it the less pressure is required to control it. And controlling it with less pressure also means that slight variations in pressure with have a major affect on how the reed plays.
This is why the "brute force embouchure" is attractive to beginners. It seems to be more "consistent" because small variations in lip pressure doesn't have much affect.
So that's my neophyte ramble. Whether there is any truth to anything I've said here is beyond my ability to determine.
))
Just consider it food for thought, or over-rambled lip-service.