I've created a basic Sway page. However when I generate the link to add to my email signature, the link returns the page in Storyline mode and not the Design layout. Option to view and not edit is definitely clicked.
The link will only go to edit mode for people who can edit - i.e., just you. Everyone else will be sent to "play" mode. To test it out, try opening up a private window in your browser (or a different browser where you're not logged in) and open the link - you'll see it just displays the output without any edit controls.
There's a separate option in the Share menu for when you want to share a link to allow people to edit as well - and when you do that, it'll first ask them if they want to edit or just view.
Sorry about any confusion - I think we could make it a bit clearer what happens when you send a link to people.