If you take the road via Navarete yes, you will see big signs for Sto Dgo overhead on the Navarete-Santiago part of the highway, just before you enter Santiago, and you will come out near the hospital on the Santiago-Sto Dgo highway.
From Sosua you may be better off taking the Moca road, or the Nagua-Sto Dgo highway....not sure....but the new bypass will save you about 20 to 40 minutes of driving that particular route, depending on traffic.