Don't worry about sinking the boat. If you have an automatic bilge pump it will only run every few minutes. There really isn't much water that comes in around the shaft. I would unscrew the nut and measure the packing, then snug up the nut again and go buy the packing. Probably 1/4 but possible 3/16.
A small hook is handy for pulling the old packing but a small screwdriver works. The also sell a puller that screws into the packing and makes it easy. You could pull one piece out to get the correct size. It should still seal up good enough for a while with what is left.
I always had good luck with Teflon Flax. Read up a bit on how to do it. 3 pieces are generally used offsetting the cuts. I always liked to snug it up good and then back it off until it would drip a little. (not much dripping needed). Check the alignment while you are down there too.
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Currently in Long Beach, CA