What you are experiencing is not uncommon when someone has played clarinet for several years and then tried switching to saxophone. Over my teaching career of 32 years I helped scores of students who started on clarinet to successfully switch to saxophone. Here are some of the things I learned:
When clarinetists switch to the saxophone they have a tendency to:
1. Not take enough mouthpiece into the mouth
2. Tilt the mouthpiece down too far
3. Play too high on the pitch of the mouthpiece (embouchure too tight)
4. Restrict the air in the throat to create resistance
All of these whether done consciously or unconsciously are an attempt to make the saxophone feel more like what they are more familiar with. Doing one or more of the above generally results in a pinched sound, a sharp upper register, and a limited dynamic range.
Ways to remove these tendencies and substitute good playing habits are:
1. Use a mp patch and for alto draw line approx. 5/8" from the tip for where to place the top teeth. For more info see:
Bruce Pearson Teaching the Saxophone Embouchure
2. While standing erect balance the sax with both thumbs, keeping the head level, and adjust neck strap so the tip of the mp touches the curve above the chin. Then tilt the head down slightly to insert the mouthpiece into the mouth. This is the correct angle of the mouthpiece for the saxophone.
3. Play the mouthpiece and neck apart from the saxophone. Adjust the embouchure so that the pitch matches Ab Concert for the alto saxophone. This is the correct embouchure tightness for the entire range of the saxophone below altissimo.
4. The clarinet has lots of resistance when you blow. The saxophone has very little. To keep the "channel" open for the more free blowing sax, say "HAUP" when you take a breath and shape the throat like you are singing "AHH" on the lowest note you can hit. Think of blowing warm air.