The draft was revived in 1969 and ended in 1976.
A draft dodger is one who tries to avoid being drafted. People would flee to Canada or Sweden, or hide out in Mexico, or try some scam like pretend to be homosexual, starve themselves to get below weight, feign being insane, or lie about being able to type 120 words a minute so they would not put you in combat, they would put you in an office some where, say Germany. You could get a student deferment, meaning as long as you were in school, you could delay being drafted until you finished college, but then you had to go. Some people were exempt from the draft, such as anyone with an oil engineering degree, because in the United States, since our wars are about controlling oil, we do not risk people with oil-related skills in war. The Vietnam war was about controlling the oil reserves in the Paracel and Spratly Islands. As US forces fled Vietnam, Vietnam was obliged to defend the islands against the Chinese.
Elvis Presley was drafted into the military, so his career had to be put on hold. Muhammed Ali was drafted and refused to serve, so he was imprisoned.
We had to carry draft cards to if we were stopped by police or military police would could prove we were not obliged to be in the military. In those years military police and Shore Patrol (Navy MPs) also patrolled the streets of major cities, to back up the police when it can to trouble with military personnel.
Conscientious Objectors are not draft dodgers. The Selective Service i in charge of randomly selecting people to be subject to the military draft. Conscientious Objectors are not subject to the draft, so there is no draft to dodge. Some draft dodgers pretend to be conscientious objectors, but conscientious objection is fairly easy to determine, and s it is one of the least likely means to avoid the draft.
Conscientious objectors do not avoid the draft, because they are not subject to the draft.
The way the draft worked is they pull birthdays of a given year out of a drum (like a lottery) and the sequence of birthdays pulled are assigned numbers 1-365. If May 10 is the first day pulled out, your draft number is 1. This continues for all dates. Now they pull a number between 1-365 out of one drum and a birthday out of another to make it doubly random.
Then the military decides how many people it needs that year, and a decision is made how many birthdays need to be drafted to meet the needs of the war machine. Say they decide taking the first 103 birthdays will give them enough people for the year, then announce with a birthday lottery number of 1 through 103 gets drafted into the military. Now just because you are drafted does not mean you will be assigned to combat, but just about everyone in combat was drafted to be there.
Each year they run a lottery to make available however many draftees are needed for the year.
As to timing, the way it works is the numbers they pull in say 1972 is for the people they will take in 1973. In essence at that time you got drafted at 18 and put into the military at 19. The people drafted for the war in 1972 were fighting in 1973. Now it is 19 to go into the military at 20.
In the Vietnam era draft I was 18 in 1973. You do not wait until the draft you to get out of the draft (you cannot get out of the selection process, only the draft.) I informed the Selective Service of my conscientious objector scruples in 1973, at the time I was obliged to register for the draft. In a 1973 lottery, I would be drafted in 1974 if my number was low, and as it turned out, I did get a very low number (023). Some people did nothing and hoped for the best.
A happy surprise in 1973 is the military did not take any of the drafted, but the lottery continued. And with defeat of the United States forces in Vietnam in 1975, another surprising event, the draft was ended in 1976
Presently it is an all volunteer military, in the sense you can volunteer to go in, but once you are in you cannot get out, like a gang. And those that tend to volunteer tend to be people with few other options.