Your chocolate is seizing. There's a nice discussion of exactly what's happening at the molecular level here:
What happened to my melted couverture?
The usual fix for seizing is to add more liquid, but I think you've got so much liquid in there that you can't agitate it well enough to break up the clumps. So you get lumps of seized chocolate swimming in liquid.
A better solution is to dissolve your chocolate in a small amount of boiling water, over medium temperature rather than in a double boiler. Since there is no milk in it at that point, you don't have to worry about overcooking the milk proteins. Then you can add the dairy to the hot chocolate syrup, then heat through. Because the chocolate is already dissolved, it will distribute evenly through the milk.
It shouldn't need any thickening agent. The chocolate is starchy enough on its own to thicken the dairy. (If you do like it with the starchy mouthfeel, you can add a small amount of cornstarch with the cold milk.)
The recipe I use runs something like this:
Put 1 ounce of bittersweet chocolate, 1 tbsp sugar, and a quarter cup of water in a small pot
Cook over medium heat, stirring until it becomes a thickened syrup
Add 8 ounces milk
Heat through, stirring
Add 1/2 tsp vanilla extract
I like to whip it with a stick blender. You can also use half-and-half for a richer result. You may want more sugar; you can add that at the table.
Similar recipes are also made in a double boiler, e.g. www.food.com/recipe/angelina-s-hot-chocolate-334398. I don't think the double boiler is necessary.
Note: it's better to use eating chocolate rather than unsweetened. Unsweetened chocolate isn't as thoroughly "conched", since it's usually used for cakes, where that doesn't matter as much. In a drink it'll be slightly gritty. (I'm also told that they use lower-quality beans, but I haven't done a taste test.)