We kept our main bank and a 2nd online bank we had for our investments. Just switched the address from our new home at the UPS Store.
All checks are either Direct Deposit or if our mail forwarder sends us a paper one, I can deposit it on my iPhone/iPad app. Really simple.
All Bank statements, CC and utility bills are paperless. I pay everything either automatically online or pay as we charge.
Best thing about paperless, the only mail we seem to get anymore is from charities sending us new address stickers or Greeting cards.