How can I avoid chipped edges on glass etching with a Dremel?
It's hard to diagnose without seeing the bit you used, but here are my ideas:
Orientation
The line in the center of your image has only one chipped edge on one side. That implies that the problem either arises at the side where the tool starts touching the glass or leaves the glass.
If you turn the glass (or the tool) so that the rotation of the bit is parallel to the line you're etching, the chipping might still occur, but it won't be visible because it'll be part of the line.
Grain
The bit you're using might be too coarse for glass. Of course the bit needs some grain to be able to etch glass at all, which would be visible as bumps under the microscope. When one such bump touches the glass, it breaks tiny parts of it away. High grain bits have a lot of tiny bumps, which break the glass away evenly like a chain saw would cut a block of ice evenly. With lower grain these bumps get bigger and fewer, chipping away chunks of glass like an ice pick would chip a block of ice.
Uneven rotation
I've made the unfortunate experience that bits sometimes won't be centered properly in the collet. You can have the same effect if the shaft of the bit is slightly bent or if the stone of the tool (if it has one) isn't centered. As a result, instead of a constant grinding you get a tool that jumps over the surface like a miniature jackhammer.