I think that's mostly a "you" problem. Connectivity gives you an option, not a requirement. Rather than being dictated by external forces now it's on you to make decisions that work for you. If constant connectivity is causing problems for you then it's on you to learn to silence your phone or close your email client or what have you and to simply not care about the resulting missed calls or delayed messages.
In general my phone doesn't ring. I respond to written messages every few hours. That works for me but it might not work for you.
I wish it was always my choice. The introduction of internet to the underground in London, has allowed others the choice of holding their conversations loudly within close proximity to me. I have no wish to know why you have fell out with your best friend, I'd prefer silence.