I usually use a combined footing if the columns are that close. I've actually used combined footings when the columns were much further apart than that.
I have a spread sheet that I wrote that covers the design of these footings making them very ease to design. Maybe that's why I like to use combined footings.
I usually just check the bearing pressure under the footing. Bearing should be related to this pressure. Whether or not you want to allow the resultant to end up outside the middle 1/3 of the footing is up to you.
Also, the spreadsheet I wrote does not allow for shears or moments to be applied in two directions.
Once you have you bearing pressure distribution that you show in your attachment. Just draw the FBD with the column loads and the bearing pressure from that you can draw your shear and moment diagrams.
Then design for the critical negative and positive moments. Check the shear and that is it.
As ash060 states, the 0.6D case can cause some problems. Many engineers will tell you to make sure you have 100% bearing under the footing. This has caused problems for me with the design of footings in the past. It would be your decision as the EOR (or your boss) of what you are comfortable with.