Wasn't Vietnam winding way down by then?  I thought they started the draft earlier than that?

Huh?  

The Vietnam era draft ran 1969-1973.

since      /sɪns/ Pronunciation Key - Show Spelled Pronunciation[sins] Pronunciation Key - Show IPA Pronunciation
–adverb
1. from then till now (often prec. by ever): He was elected in 1978 and has been president ever since.
2. between a particular past time and the present; subsequently: She at first refused, but has since consented.
3. ago; before now: long since.
–preposition
4. continuously from or counting from: It has been warm since noon.
5. between a past time or event and the present: There have been many changes since the war.
–conjunction
6. in the period following the time when: He has written once since he left.
7. continuously from or counting from the time when: He has been busy since he came.
8. because; inasmuch as: Since you're already here, you might as well stay.
[Origin: 1400–50; late ME syns, sinnes (adv.) thereupon, afterwards, ME sithenes (adv. and conj.) afterwards, from (the specified time), because, equiv. to sithen after that, since (OE siththan, orig. sīth thām after that; see sith) + -es -s1]

—Synonyms 8. See because.
—Usage note 8. See as1.
Dictionary.com Unabridged (v 1.1)
Based on the Random House Unabridged Dictionary, © Random House, Inc. 2006.

You do realize that 'since 1973' means after that date?

73 might have been the last year that anybody was drafted into the military.  It had to be very few as Vietnam was winding down in terms of US involvement by then.  Males have been required to register at age 18 for a while.  Not sure about your question...

dvo  

Last Lottery was in Feb 72, and no one from that lottery was drafted.