Alright. Just download a stock Rom of your choice from the forum here, put it on your SD card and flash away. It should boot right up bud.
The reason CM probably didn't work for you is because anything below a CM10.2 nightly won't work on the new galaxy s3s because of some issues. If you want a cm ROM, try a new nightly. All in all, just put a ROM (I would say stock based so you can use your phone) on your SD card if you have one or you could even side load one from computer.
Sprint Galaxy S3
Stock - Whiplashh Rom - MIUI - Wicked Sensations
Sprint Evo 4G "OG"
Stock - Cyanogenmod 7 - MIUI - Swagged Out Stock
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Sprint Evo 4G "OG" - Stock - Miui - Cyanogenmod 7
Sprint Galaxy S3 (Fixed!) - Stock - Miui v5 - Wicked Sensations - Cyanogenmod 10.1.3
Sprint Optimus G "LGOG" - Stock - Stock KitKat Port (Android 4.4) - Cyanogenmod 10.2
Sprint Optimus S -Stock - ICS AOKP Milestone 4
Sprint Evo 4G Shift (Dead) - Stock - Miui - Cyanogenmod 7
Google Nexus 7 (2013 Edition) - Stock