Stuttering is worsened by a magnesium deficiency and so is myotonia. Magnesium is depleted by exercise so it's possible that's why you're seeing a correlation between the two.
I would suggest buying some magnesium citrate powder at the health food store. Start with a small dose twice a day and see if that helps. Too much will cause loose stools so you have to increase it to your tolerance level gradually.
Another sign of magnesium deficiency is hiccups :-)
I don't have a stutter but sometimes the MC causes me to trip my words up, or slows me down when speaking and so I don't always pull off the smoothest word delivery. This can make me seem more "nerdy" than I actually am, or less confident than I actually am, which can be frustrating, so I can relate.
I saw a device on a documentary once for people with severe stutters. It fed back what the person was saying, into their ear, which distracted their brain, causing them to speak a little slower but without the stutter.
On another note, I used to have a handy man who had a severe stutter, and still insisted on every other word being a four letter one. He had a lot of perseverance :-)