I also live in a county where a sherrifs office has patrol cars and K9s in addition to many other divisions.
Gooochy in my county. The sherrifs department has the townships and 3 cities that they cover in the far southern part. They cover actually a very large area. But they do the exact same thing as patrol officers do down there. They respond to the same calls, traffic stops, and yes sometimes respond county wide m/a with the K9 officer. The sherrifs office does have a dvision that handles the court calls, evictions.... Etc. Obviously everywhere is different.
And volunteer. I do see what you are asking. However I am not sure if they would of not. Originally the sherrifs units were not even designed to be sherrifs. The devs made them and they were looked at as supervisors. When we first got them they couldn't even transport. I mean never say never however. I wouldn't hold your breath on this one.
Not sure about every state but Minnesota. Even state patrol has K9's, Swat Teams, negotiators. And they cover parts or the state as the only law enforcement around. They are contracted.
Hopefully that helps.