Yes, you remember correctly but I don't know what the difference is between volunteering and enlisting (or did you mean to say "drafted" instead of volunteering).
I "volunteered" in the same sense that others like GWB volunteered for the National Guard. I joined the Navy because I expected to be drafted after I dropped out of college and lost my deferment. Only years later did I find out my number would have never come up. Most of us who joined the Navy or Air Force did so to avoid being drafted and ending up in the Army where the odds of ending up as a grunt fighting in the jungle were much higher.
Never even thought about going to Vietnam at the time. Actually, I didn't even know I was going there until the day before I arrived in DaNang which, needless to say, was quite a shock. According to my orders I was supposed to be going to a Technical Research Ship (AGTR-1) which sounded really interesting at the time. I spent a few months on that ship before it got decommissioned due to the U.S.S. Pueblo incident in N. Korea and then got transferred in-country where I spent the rest of the time in the mobile wolverine force, leaving just before the Cambodian invasion.