If it's urgent you should still call first to confirm that the person is available and able to deal with what you need them to deal with and that what you're sending them is what they need.
Anything else risks wasting a bunch of people's time going back and forth which if it's genuinely urgent (it usually isn't) everyone should want to avoid.
If someone's likely to call you to follow up on an "important" email that isn't important, turning off email doesn't help. You'll still get a vibrate/ring, look down at your phone, and decide whether or not to pick it up -- exactly the kind of distraction we're trying to avoid here.
The problem with the call-me-with-only-urgent-stuff approach is the Boy Who Cried Wolf problem: if people are constantly calling you (because they know they're not going to get a reply until later in the day, and they need it <whine>noowwwwwwwww</whine>, you'll put your phone on DND/silent. But now you can't respond to actual emergencies.
In the first case, you must take action to appease them, whether replying to the email or taking their call. (Or you could script your interaction with them; an automatic email reply or such. But that's suboptimal too). It's outside of your control.
In the second case, they aren't able to distinguish what they consider to be emergencies from what are actual emergencies; this, too, is out of your control.
I honestly don't know how to deal with these. Thoughts?
My solution is as I described. I make it clear that I am unhappy. I make it clear why. And if they refuse to listen, I arrange my work so that they have no excuse for trying to ruin my day.
Anything else risks wasting a bunch of people's time going back and forth which if it's genuinely urgent (it usually isn't) everyone should want to avoid.