Another fool-proof method (no need to worry about having a specific backup app) is to grab the game's data directly.
The data for all user-installed apps can be found in
/data/data/*package name*. This will be all settings, saves, and anything else. Just browse into there and copy the relevant directories to somewhere safe and put them back when you're done flashing and have reinstalled the app and run it at least once.
Yet another way, which will work without root, is you do the same as above, but using adb. Turn on USB debugging, connect up to your computer, and run
adb pull /data/data/*package name* C:\where\you\want\the\files.
Then once you're done with your ROM flashing and have reinstalled the apps, do
adb push C:\where\you\put\the\files /data/data/*package name*, or if you now have root with your custom ROM, just copy the directories you pulled to the SD card and put them back in
/data/data on the device.