I won't expand on the technical issues, they've been covered in enough detail.
I will suggest that if you have a "freeze flow" you can shut off the valve for a minute or two and then turn it back on. Generally that will be enough time for the ice that formed to thaw, so if you can bail out to your buddies octo, you should be able to go back on your reg to finish the dive. That in many cases will leave you in warmer water for the safety stop with the ability to finish the stop, inflate your BC etc in a normal manner. Something to consider if you can reach your valve or have an on the ball buddy and minimize the gas loss.
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If you are in a cave or wreck, the first priority, believe it or not, is not to stop the freeflow, but rather to get on the line and then deal with the reg. The extra second or two that takes will be a lot more gas efficient that the time it might take to locate the line if the viz gets blown. Even if you don't contact the silt and stir things up while shutting down the valve, switching regs, etc, the high gas flow will cause percolation and potentially water movement in a small space that can easily blow the viz. The same is true if you are donating a reg, donate the reg and get on the line immediately.