According to older grammar, one should say "I shall" when referring to the future: "I shall visit you next week."
One should use "I will" when expressing determination: "I do not like him. So I won't talk to him if we accidentally meet at the party."
Here in the United States in 2021, everyone uses "won't" for the future, too. If you say "shan't," people will look at you funny. They might think that you are joking.
Personally, I like "shan't" because I am very old. (I shan't tell you my age.)