Actually, on the Tokaido Shinkansen,
there is one regular westbound train starting not from Tokyo but from Shinagawa,
which is Nozomi 99 bound for Hakata
(Shinagawa 6:00 -- 7:28 Nagoya 7:29 -- 8:16 Shin-Osaka).
During the 10 days of Golden Week 2019 (from April 27 to May 6),
Nozomi 151 is available from Tokyo Station
(Tokyo 6:06 -- 7:42 Nagoya 7:44 -- 8:33 Shin-Osaka).
It may be good news to you
that Nozomi 99 / Nozomi 151 currently still have some seats to be reserved,
in the Ordinary Class car(s), from Shinagawa / Tokyo until Nagoya;
paying 930 yen more, you can have a reserved seat, though partially until Nagoya.
For your ride with your seat reserved,
you in advance specify the first station (Tokyo or Shinagawa for your case),
which is indicated on your ticket.
In Shinagawa Station,
there is only one Shinkansen zone;
you can just follow signs for the Shinkansen.
https://global.jr-central.co.jp/en/info/station/_img/shinagawa.jpgWhen you choose Tokyo Station,
there are two separate Shinkansen zones with gates of same or similar names.
Please pay attention
to the names of the Shinkansen lines and/or the colors of the Shinkansen pictograms.
https://global.jr-central.co.jp/en/info/station/_img/tokyo.jpgThe Tokaido Shinkansen zone (, which is managed by JR-Central,)
is indicated with blue-and-white Shinkansen pictograms,
like the one seen in the photo of the next link.
https://www.japan-guide.com/g3/2018_how_sign.jpgThe other Shinkansen zone (, which is managed by JR-East,)
is indicated with green-and-white Shinkansen pictograms.