The Earth orbits the Sun in a counter-clockwise direction. Covering $2 \pi \times 150\times 10^6$ million kilometers in $365.25 \times 24 \times 3600$ seconds, or 29.9 km/sec.
If you wanted to leave the Earth in a clockwise heliocentric orbit you'd have to accelerate Dawn by an extra 60 km/sec to do that, which is unfeasible with current rockets.
Likewise approaching Ceres you'd also be going the wrong direction to try to enter into orbit, you'd face the problem of Ceres moving counterclockwise at 17.9 km/sec while Dawn would be moving roughly the same speed clockwise, so another 35.8 kilometers per second relative motion that you would have to "dispose of" somehow.
As this answer points out, Dawn used electric propulsion which can only slowly change velocity. If you are in a heliocentric orbit and tried to spend many years decelerating to a dead stop and slowly re-accelerating the other direction, in the mean time you'd fall towards (but probably not into) the Sun, if it were possible at all (it would probably run out of propellant).
Dawn was not built to withstand a close pass with the Sun, and so it would not have survived.
Launches often try to use every last bit of "free" velocity from Earth's motion, not only by the deep-space launches orbiting the Sun in a prograde direction (the same direction the Earth and all other planets rotate) but by first launching into orbits around the Earth in a prograde direction, taking advantage of the Earths rotational velocity around its axis.