Quitting Braid causes dual monitors to switch to mirror mode
I don't have braid myself but have found this issue else where. What I can only suggest is two workarounds:
1) run in windowed mode, as detailed in comments above, which sounds like it works.
2) create a new launch script as detailed on the WINEHQ page linked above, to use xrandr after braid quits to return the correct resolution./opt/braid/braid && xrandr --output DVI-0 --mode 1024x768 && xrandr --output DVI-1 --mode 1440x900
edit: changed suggested xrandr command to match OP output