You are correct. And before that they were under the Dept. of Transportation (DOT).
That being said; I don't think (or know of) there having been any change in having Coast Guard recruits go through Navy basic training, before being sent to their duty station.
When I was in Great Lakes, myself, I remember the Coast Guard recruits being looked down upon by the others. It didn't take me too long to realize the error of that way of thinking....
In those days we learned;
N. never
A. again
V. volunteer
Y. yourself
Nowadays, I'm even considering reenlisting, but I won't.
When I was in the Navy, we laughed at the Coasties, too. But I changed my way of thinking about them. These guys put themselves in harms way an a regular basis.
Thank you for your service - that's encouraging to hear.