It seems to me not that you're having problems with how the Americans do it but with proper punctuation of dialogue in general. Which is common, and learn-able, so listen up.
Travis crumpled the map into his hip pocket, pulled the walky-talky off of his belt and pressed the talk button. “It’s not even here!” This needs a period after the word button not to be American but because that's the end of the sentence. Pressing the talk button is not a synonym for said and therefore can't be treated as a dialogue tag/attribute.
Travis crumpled the map into his hip pocket, pulled the walky-talky off of his belt and shouted, “It’s not even here!” Note the comma, because shouted is an appropriate dialogue tag.
The same thing goes for your second question. Travis pressed the button to talk, but Kaylin leaned in. “I bet there’s no such a thing as GPS treasure hunting.” You'd use a period after leaned in because leaning in isn't a synonym for said and can't be a dialogue tag.
Travis pressed the button to talk, but Kaylin muttered, “I bet there’s no such a thing as GPS treasure hunting.” This time a comma is needed, because muttered is a dialogue tag.
Lots of people have trouble with this. They have characters smile, laugh, or nod lines of dialogue all the time. To self-test, could the "tag" word or phrase (which makes the comma separating it from the dialogue correct) be replaced with said and still make sense? If it can, comma. If it can't, period.
Maryn, hoping that made sense