I figured it out. The My Documents folder containing my Dolphin save data somehow ended up in the My Pictures folder. Dolphin made a new folder called "documents" and created another GameCube memory card there. I assume that's the card the games saw as corrupted and wanted to format. When I moved the My Documents folder to its original location I could access my original save data again.
PC Specs: Pentium G3258 (stock speed), MSI Radeon R7 360 2GB GDDR5 OC, 8GB DDR3, Windows 7
