If my 19-year-old son joined the Navy in March and is currently in A-School, do I claim him as a dependent since his residency is still here at home?

No, Military school does not count as being a student when it comes to claiming a child as a dependent. The training in the Navy is considered to be as an employee, not as a student. 

In order for to be considered a full time student, he would have to be at a technical school for something unrelated, that was not part of his military service, then this would count as a full time student.  

Thanks Vanessa. Follow up. Do I still answer "Did [my son] live with you for the whole year?" as a yes since his residency is still legally here?

Yes, that is correct. You would report your son as living with you for the entire year.