Its final orbit will only be 375 km above Ceres, but you have to give it time.
Dawn is powered by xenon ion engines, which are extremely efficient, but very weak. The usual comparison is that they are pushing the craft forwards about as much as a sheet of paper pushes down on your hand. Their advantage is that they can do this for a very long time. This is why Dawn has been able to visit Vesta and Ceres.
A probe with conventional chemical rocket engines can change its speed very quickly, but it only has fuel to fire a few times, briefly. What has been done in the past is that probes have been carefully aimed at their destinations, and then when they arrive, they swing in really close to the planet and fire their engines when grazing past at the closest point. This is the best way to slow down. By doing that over a few passes, it can be done with only brief firings and that saves on fuel. This is called an Oberth maneuver.
Dawn doesn't have to fire engines at the closest approach to Ceres. Its engines just keep firing, and firing until finally it slows down enough to orbit much closer. In fact the word 'firing' doesn't really apply. The engines have been running, continuously, for thousands of days. (Well, there's three, and they switch which is on periodically, and there were some periods of coasting.)
Braking this way takes weeks. It has already been braking for weeks as it got closer to Ceres. Then at first it was captured only into a highly elliptical orbit whose far point was very far away from Ceres. Still it runs its engines over a large portion of its orbit rather than applying much greater force briefly at the orbit's lowest point, and thus the Oberth effect is much less important in its maneuvers.
But because the engines are so very efficient, still less fuel is needed that if chemical rockets had been used. The luxury of having engines that run on and on may also have helped mitigate the problems of dealing with the failed reaction wheels, and also the (temporary) failure of an entire engine did not doom the mission.
(Note - Probes like Cassini and Voyager managed to visit several destinations by making extremely clever use of gravity assists as well, carefully entering into the gravity well of a planet or moon in just the right way so that they got swung around them and thrown out again in just the right direction to proceed on to the next destination. It is only because they were set up to do this that they managed to make it to several awesome destinations. But they wouldn't have been able to slingshot very much with these low gravity destinations.)