What is different?

I have no real logic for why this is but to me, “should” sounds like you’re asking for advice versus “shall”, which sounds like you’re asking to do a favor for someone. “Should I send the gift to her”: Feels uncertain, like you’re asking someone if sending a gift is a good idea or not. “Shall I send the gift to her”: Do you need me to send this gift to her? Sounds like you’re doing a favor for someone. At the end of the day, both are understandable. I definitely view “shall” as more formal; it’s also more common to say in the UK versus America (where it would be seen as overly formal, but not super strange).

Should is a suggestion - You should consider trying the fish. Shall is a command - You shall eat the fish.

I think "should" is more common. "Shall" is more correct, and more formal. However, it is *extremely* common to say "should we go" in place of "shall we go" (for example). There is a difference in the meaning of the words; "Should" implies doubts. If I say, "I should go to work now", I probably mean that I ought to, but I can't be bothered. I'm reluctant. I might have another cup of tea first. "Shall" implies certainty. If I say, "I shall go to work now", I'm about to set off. However, when used as a question, the line between them blurs, and "should" is a more common word.