The crime reported in the newspapers as "f. and a." was "Fornication and Adultery", as in this case from 1922:
The penalty handed down to Oakey Ferguson in this case of "one year on road" meant serving a year working in a NC road gang.
As I understand it, Fornication and Adultery remains an offence (actually a Class 2 misdemeanour) on the statute books in North Carolina.
[If you happen to live in North Carolina today, and you are thinking about dating before you are legally divorced, basic legal advice is available from a number of websites.]