Beginning Doubler Seeks Advice

For flatness on high notes, may I suggest that you need significantly more air pressure, and the lip support to "contain" it - a lot more than for sax.

Also, be aware that playing in an extra cold or hot environment really plays havoc with clarinet tuning.

For the A-B, the short answer is practice, practice, practice. It is possibly the most clumsy of all clarinet fingerings, but we master it, because we have to!

If it is a trill, where tuning and getting to the next note after these is not too critical, then use one of the four side keys - i.e second down from the top, to change A to B.

Otherwise, there are several fingerings for A which leave most of the B fingers down, so it is easier to change from one to the other.
See http://www.woodwind.org/clarinet/Study/FingeringCharts/bbfinger.html

There are many other combinations of fingers left on for A that work. Some work for some clarinets better than for others. For example on my clarinet, for the A, I often leave down left middle and ring, and right index and middle fingers. This gives more resonance for the A, often a pitch I prefer (depending on volume and air temperature) and an easy transition to B. (I leave these down also for the throat G#)

(Likewise, for the throat Bb, I leave down left middle and ring, and right index, middle and ring.)

For me, the differences in embouchures are as follows:

Sax needs a significant section of reed in contact with the lower lip. Clarinet needs a much shorter section of the reed in contact with the lower lip.

Therefore for clarinet, the lower lip must have a lot less less bulk.

So:

Sax: Bunch up the lips as in whistling (but the mouth open enough to accept the mouthpiece.) Sit the reed/mouthpiece on the thick cushion of lower lip. Then close the lower jaw until the upper teeth are on the mouthpiece, and close the lips around the mouthpiece like a rubber band.

Most of the lip tissue that the reed contacts is the red part that is normally wet (and soft) from being normally inside the mouth.

Clarinet: Smile (with teeth apart) sufficiently to take all wrinkles out of the lower lip, and stretch it thinner, and allow a small overflow of lip over the lower teeth. While the lip has been thus distorted, place reed/mouthpiece on the thin strip of lower lip that covers the lower teeth. Then close the lower jaw until the upper teeth are on the mouthpiece, and close the lips around the mouthpiece like a rubber band.

Most of the lip tissue that the reed contacts is the red part that is normally dry(and not quite so soft) from being normally outside the mouth.

Of course, there are many different approaches. And we each have differences in our teeth and lips.

I think that for clarinet I also have the back of my tongue raised a little more towards my hard palate.

Disclaimer: I am a mere analytical, experienced, self-taught amateur player.

Not to be argumentative here folks, but look at the title of the original post asking for help.

Beginning Doubler Seeks Advice

Note the word "Beginning". To advise any beginning player on clarinet to adopt a "double lip" embouchure right from the start is not helpful advice in my opinion. True the "double lip" is used successfully by a small minority of accomplished players on both sax and clarinet, but most of those players switched to that embouchure after years of playing and mastering their instrument. From what I have read, most of them describe how hard they had to work in order to make the change. Getz tried it and then went back to his original embouchure.

If the "double lip" embouchure is so easy and advantageous, why then aren't those players in the majority instead of the other way around. I have heard it said that John Coltrane was not a great player because of his "double lip" embouchure, but in spite of it. For any experienced player to "grow into" the "double lip" embouchure is great if it works for them, put please don't suggest that for a player who is just learning to play the instrument.

John