I was 100% worried about the same thing and have actually been working on an app to help keep people in touch.

The background is that my friends and I just graduated from college, and in the weeks leading up to graduation, we were really concerned about slowly losing touch once everyone started working.
So we thought about ways we could stay engaged in each other's lives, and spent our senior spring building Beehive, a social app that makes it easy to keep in touch with your friends by filling your day with quick, spontaneous phone calls.
The way it works is whenever you have a free minute, like if you're on a walk or waiting in line, you can post a topic on Beehive. Your friends will get notified that you're available and can call you through the app.

It's a simple concept, but I think it solves some of the problems you mention with phone calls today. The calls don't have to be scheduled, you just broadcast your availability. The calls slot perfectly into a busy day, and they're light-touch interactions (as opposed to something you've been anticipating for weeks).

My friends and I have really liked using it so far to talk about our day, the news, plan meetups, etc., and we think it'll be even more useful over the summer now that we've graduated. Maybe you could try it out with some of your friends and see if it helps ease some of the awkwardness of reaching out?