Unfortunately California is one of the special states where they have specifically and explicitly separated Civil & Structural. More states are going or have already gone this route, especially on the West coast where earthquakes are prevalent. Seismic design is not adequately covered in the Civil-Structural PE exam.
Personally, I fall into similar shoes as I took the Civil PE with structural emphasis back in 2010 before Alaska separated Civil & Structural. We referred to ourselves as Structural engineers prior to the regulation change. Shortly thereafter, Alaska changed the regulations to be more in line with several of the other West coast states and separated Structural engineering from the Civil block. Engineers who were practicing Structural engineering could "grandfather" in under the new regulations for a set time period and receive a SE license in addition to their Civil license. For those of us that were grandfathered in, our Structural license is only applicable to Alaska and cannot be used for reciprocity in any other state. To gain reciprocity, we must take the 16 hour Structural Exam.
You will be unable to go after the Structural specific work in California (and several other states) unless it is federal or you have received a California SE license. Some of the states with SE licensure require a civil license for 2 years in the state prior to being eligible for the Structural exam.