If you have room to run both and it's not disruptive to your tank, keep both filters going. It's always nice to have a spare to startup another tank.
In regards to cleaning fluval/hagen aqua clear filters, yes, swishing, squeezing the sponge out in tank water is the preferred method. I sometimes rinse my AC sponges in flowing tap water too. I swish the bag of bio media in tank water to take off a excessive built-up if need be.
You're supposed to replace the carbon bag monthly. Every two months is probably fine depending on the bio load. On my planted 10g, I don't use the carbon and just double up on the sponges.