I have only seen accounts of people mentioned that both Sun and Jin are without rings. If you can provide a screen cap with Sun wearing one from LA X, I'd like to see it.
And about the money: First, Sun did not have money packed when she was planning to escape Jin during their "vacation" to LA where Jin was delivering the watch. (s1e06)
However, if you remember (s3e18), Sun was blackmailed by Jin's mother than she will tell everyone that Jin's mother was a prostitue. Sun gets to $100,000 from her father in exchange for Jin's work for him. So perhaps in this alternate reality, Sun obtains the money from her father, Jin works for Mr. Paik, but for some reason they don't give it to Jin's mom. Maybe Sun kills her instead. She did threaten her life at the time time of the money exchange in the original plot. Who knows.