when you're in the menu, one option saves and return to the main menu but below it there is one to give up and delete the save. just starting a new character will also overwrite the save.
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I believe this was changed recently :)
Deleting the save was made harder based on feedback from some players that having the delete save button accessible in the pause menu allowed muscle memory to do bad things. So now you have to explicitly overwrite the save (with a warning) when starting a new game.