Your instructions worked like a champ... Though I had some difficulty laying hands on a recovery image to use...
And I ran into a few problems because I was running the UVALH2 update... so most thing's were fcing after flashing with root... but after getting a recovery on, I was able to flash a stock LH2 image with root over top of my system via the recovery, and I'm cooking with gas... root, on lh2 without any counter increment.
I guess I should update the OP to say that if you do this, you should probably also do a factory reset. Thanks for the feedback.
For recovery images, your best bets are to either use Goo Manager (preferred for our Galaxy S3s due to the IMEI issue) to install TWRP or ROM Manager to install CWM. If you want to manually install CWM (or want CWM Touch for free), you can always get them from here
EDIT: Updated the OP with info to help others from having both of your problems (finding a Recovery and bootloops/FCs due to "downgrading" the OS image).