I am starting alto sax but play clarinet.
On the clarinet I am not too bad in the upper registers and have improved, especially accuracy and tone, by using a harder reed - worth a try.
Must be honest, as I am new to the sax, I find the upper register pretty easy, my difficulties lie in the lower notes on the lower register.
Another comment not related to this but related to new sax playing, I find that the semitones on the sax are a lot less intuitive than on the clarinet but I suppose that is because I am used to the clarinet.