With iOS 6, my wife and I did encounter the problem where "My Info" under "Mail, Contacts, Calendars" and "Siri" wouldn't stick and kept changing the way you describe. I worked around that by removing both our contacts from iCloud. I then transfered those two contacts to both our phones using iTunes. The end result was that each phone had two categories under contacts, "iCloud" and "From My Mac" (with just two contacts). Then setting "My Info" to a "From My Mac" contact made it all work. The latest iTunes says something to the effect of "you are using iCloud contacts, so no Mac contacts for you". With iOS 7, setting "My Info" to a contact from a secondary iCloud account works instead.
About the AirDrop problem,
I turned off iMessage and and FaceTime on all our iOS devices and computers. I have deleted my contact from everwhere possible. AirDrop from my wife's phone still has my name on it, instead of hers.
I am biting the bullet and switching my wife's phone and computer to her own iCloud account. Common purchases are easy. Contacts, Calendars, Reminders, Notes are easily shared by adding each other's iCloud account as a secondary account under "Mail, Contacts, Calendars". I don't like the loss of a single Photo Stream. Having our own Passbook and Documents/Data is probably alright. I am not too excited about giving up common iCloud Safari tabs, but finally having our own separately organized bookmarks will be nice. We each get our own 5GB.
Apple's philosophy appears to be "a separate iCloud account for every person". I don't think sharing an iCloud account will get any easier with upcoming features like iCloud Keychain, which is supposed to hold credit cards and passwords.
The ability to easily share stuff amongst family members is important to many of us. I can only hope that Apple recognizes this and makes things simple in the future.