The lip still forms a cushion between the teeth and the reed, but it's more relaxed and  the corners of the mouth are low.  That allows the reed to vibrate more freely, allowing better tone and response.  Look at some close-up pictures of big name players in action and you will see what I mean.  The next step is to learn to control intonation using your oral cavity and tongue position rather than by lip pressure.  This is all a pretty major rebuild of your current technique that will require some effort starting with long tone practice, but it will be worth it.  Not too long ago I had to rebuild my technique to get out of the habit of relying too much on lip pressure for control, which was also leading to biting and lip fatigue.  It's nice to have more endurance and not be gnawing on mouthpieces any more.  So develop good habits and the possibilities will open up for you.

I also switched from clarinet, but I went from clarinet to tenor.  I quickly learned that the over-the-teeth embouchure destroyed  the tone and response below about G1.  Maybe that would have been less apparent had I switched to alto.

An instructor who insists on the clarinet as a foundation for saxophone is suspect in my book.  There are some voicing and intonation skills required for saxophone playing that go beyond those required for a clarinet.  It's a myth saxophones are easier; they require some different skills to play well that might not be apparent to someone who hasn't really taken the trouble to understand them.  It really sounds to me like you had some quick fix, path of least resistance instruction that is now working to your detriment.