FFVI: Divergent Paths

What is the best way for me to update from 1.0 to 1.1 without losing my progress? 

You don't want to patch DP 1.1 over DP 1.0.  You want to make a copy of a clean FF3, patch it with DP 1.1 alone, and then rename it to whatever your ROM of the previous version was named.  That will keep the save data pointed in the same place.  Or... you can make a copy of your save data from the emulator folder and name it to match the new ROM.