Better word than "district"
You could try the word "province":
an administrative division or unit of a country.
Also: "realm" and "domain".
County, shire, range, division, territory, parish, march, zone, partition, ward, constituency, tract, suzerainty, borough, vale, colony, riding, municipality, realm, duchy, fiefdom or earldom.
Though the last few are rather more dependent on the governance than most of the others
How about these:
Precinct
or
Prefecture
?
Another possibility would be canton.