I am unable to find an API that will return the district_court OCD division for an address. I have tried the Google Civic Info API (specifically https://developers.google.com/apis-explorer/#p/civicinfo/v2/civicinfo.representatives.representativeInfoByAddress) and look through the Open States API.
I'm trying to do this for North Carolina addresses. Here is the map of current districts: http://www.nccourts.org/Courts/Trial/District/Documents/DistrictCourtmap15.pdf