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Look up nichrome wire transformer calculator on the internet, it will let you calculate the power supply based on the thickness and length of the wire your using
Your wire is still too hot, look up nichrome transformer calculator online to figure out the right amperage for the length, like for a 12 inch wire of 30ish gauge wire you shouldn't be putting more than 1.6 amps through it. For your 10V DC power whats the Amps (it should be 16 watts, if its more the wire is too hot)
Also for the budget oriented people, you can salvage like 10m worth of Nichrome wire out of a broken hair dryer (just take it apart, its the silvery wire around the edge of the airflow tube, just uncrinkle in with a bolt thread, there is usually 1-3 different gauges of it inside and its designed to take the heat. also there is usually lots of mica heat board in there too , good to act as insulated guides for the wire as well
If you do a running stitch it will make a good finishing stitch, but invest in a thimble to push the needle through as the webbing can be tough, also use a heavier gauge needle so it dosn't bend , like a denimn or small leatherworking needle