That’s what my father did (USN). Didn’t stop the local draft board from trying to draft him and his then wife had a surprisingly hard convincing them that he was on an aircraft carrier in the middle of the Pacific.
Both my uncles (mom’s side) manage to avoid the draft in different ways. One had a bunch of metal pins in his leg as a result of breaking the same leg twice during the same school year, the other convinced the draft board that he was a transvestite with TB.
Wanna bet?
The standard tour of duty in the Army for Vietnam was exactly one year, or the remaining time left in your service if shorter.
Of course, you could voluntarily extend the in-country tour, for any reason. One reason might be to get additional R&R and stateside leave time, which was handed out liberally for troops who extended. I had a good friend who used the system to get 2 out-of-country R&Rs per year, 2 trips back to the US for 45 days each per year, and some in-country R&Rs, too. He did this for 6 years before the Army got tired of the game and transferred him to Kansas.
Kansas was so boring he quit the service when his time was up.
I know another soldier who thought he was safe from being transferred to Vietnam because there were only 8 months left in his total (drafted) tour time. He was wrong, and was sent to Vietnam for 8 months. Upon his arrival, he was shorter than 1/3 of the people in our organization.
I extended my Vietnam tour for 42 days in order to benefit from the early-discharge option. If you returned back to the states with less than 5 months active duty left, you could be discharged immediately. I could have done that with 41 days, but I didn’t trust the Army to count, so I played it safe.
I served in Monterey and West Texas and say the same thing. Sometimes I question the wisdom of saving Texas 
For the record, when drafted you were drafted into the “Armed Forces of the United States,” not by any particular branch. For obvious reasons, the Navy and Air Force never had to avail themselves of the draft. As to the likelihood of being sent to a particular branch, I can only tell you of my experience. I was drafted in May of 1969, one of 113 reporting to AFEES in Chicago on May 13. The Marines needed two, one of which turned out to be me.
Indeed, some were successful at enlisting in the Navy or Air Force to avoid being drafted. However, those entering in the Navy and trained as Corpsmen, stood a pretty good chance of being attached to a Marine unit. And those guys had it as bad, if not worse, than most others. They did one hell of a job under very trying circumstances.
My lieutenant tried to talk me into extending in country, saying I’d be sure to make E-5. I told him no fuckin’ way. Then he says: “Well, I could probably get you involuntarily extended.” I asked him if he planned to let me take stateside leave in that case, to which he replied “Yes.” I told him he’d never see me again. I made E-5 within three months of leaving that shithole.
My brother was hit with if he extended he would be made an E-5. His standard answer was “look I am not stupid enough to become a lifer so I do not need the E-5.”
I had the same kind of thing happen to me. 3 months after getting my thirds ticket and an inactive Naval Reserve commission as an Ensign, I was clasified 1A. I filed with the draft board for a change of classification, I included a copy of my comission. I recieved a letter back stating that my request was denied because and the box not sufficient reason was checked.
I went into the draft board office in Salinas and started by asking to see “the idiot who signed this letter.” I will keep it short but I got a letter saying that my case was being reviewed, but I waas never notified of a reclassification.
If drafted I planed on showing up to boot camp in uniform.
the OP might be interested in the book Chance and Circumstances. It outlines all of the ways that even draftees could avoid getting sent to Vietnam. It’s been a while since I read it, but I recall that its basic theme was that anyone who really wanted to avoid Vietnam, and was willing to put some effort into it, had a good chance of doing so.