HypCo says:
On the world map, is Heham a desert city? If not, is there any desert cities that are near-ish to Eglar? Is there a desert city that a dragon rules over?
Heham is not a desert city. It is nestled amongst vast woodland and grassy hills. Heham is also where the ancient green dragon Alikil holds court, even though his lair is elsewhere.
The Tower of the Fallen is a city close to Eglar (opposite the channel, actually), and lies on the outskirts of the Desert of the Lost. It could technically be ruled by a dragon, since it lies in the region ruled by the white dragon Sumadea. However, the governor of The Tower of the Fallen is a black dragonborn named Mesheba.
Speaking of the world map, you can consider any labeled location to be a major city or stronghold. In the past, a place like The Tower of the Fallen may have been used for a different reason, but after Caer Tan's conquest many were repurposed for other ventures. The Tower of the Fallen, for example, now serves as a port city.
There are other unlabeled locations throughout the world. These are small villages or towns that pop up wherever there's a need, or hold so little value as to be considered inconsequential to global cartographers.