These high notes are hard.
You dont say how long you have played, but they take time.
If you are patient, and do the right stuff in your practice, it will happen.
There shouldn't really be a huge movement in your lip formation. You say "move my lower lip far forward". that doesnt sound good and I think you should probably try to lose that. It may be a short term solution, but it is not good long term.
to be honest, i would not get too worried about your tongue position at this stage. that kind of stuff is a more advanced thing and best not get too confused and hung up on for the moment.
Of probable more importance is the air stream/column. Its very important that you have a concept of pushing the air through the sax. In many ways the air column is even more important in the higher ranges of the sax than anywhere else.
I like to think of the air stream moving faster in this upper range than down below.
Have a mental image of seeing the air coming out of the end of the bell. !!!!!!!!
ie you arent just blowing into the mouthpiece. You are blowing all the way through the sax.
Mental image is a big thing in regard to tone. You have to input in to your brain the right stuff.
How do you do that......... LONG NOTES. .......... Im afraid
So, pick a note that you are comfortable with and that you dont feel there are any lip contortions going on with.
for example a high D.
Play that note as a long note. Hold for 20 seconds or so. experience what is going on with regard to your breathe control, and your embouchure.
and most importantly, listen to the sound being produced. What can you do to better your tone
Do that a few times. then play the note again, but after say 5 secs, move, without tonguing , and with no break in your air stream,up to an E flat. Hold it. Experience any differences. Do this over and over
When, and only when, you can go straight to a high Eb, try the same procedure again, but trying for the high E
Etc etc
Be very patient. If it takes you a week to get a strong high E, that is fine. It isnt a race. Its a procedure. Be patient. Eventually, you are aiming for high F to high F sharp. That can take a while!!
One step at a time. IN SEMI TONES at each stage
Im not suggesting this is all of your practice time. Say 5 minutes at the start of each practice session will reap rewards.
If possible, try to avoid going for these high notes in your other practice, until the old bad habits have gone. This may take a while.
So, its long notes, going up by a semi tone at a time until secure, patience, air support, no funny stuff going on with lips.
Good luck.