I finally had some time and got my new S3 rooted using the xda guide:
and this video:
I'm now running the stock Verizon Jelly Bean rooted. I installed Titanium Backup and restored the data from my mileage app so I'm a happy camper. Maybe tomorrow I'll figure out what I can delete from my SD card. I'll have to see if there's a way to backup the mileage data a simpler way. Titanium backup seems like overkill since I can reinstall apps from the playstore.--
I'd rather have a bottle in front of me than a frontal lobotomy.