Is JR dirty tricks. I been Japan so many years, based on my experience. If you booked thru JR Rail Pass, system will know you are foreigners, it will automatic pop out all seat full.
10-15 years ago there are no such thing as online booking, ticket had to booked / purchase thru machine / JR ticket counter, Those JR ticket counter saw foreigners face, automatic put you in non-reserved seat during Golden weeks.
That time, all shinkansen still got non-reserved seat, foreigners 100% kena non reserve seat. Non reserved seat become like our KTM commuter, seat are 1st come 1st serve, if half way you walk away from your seat for pee, another person can take over your seat. My longest record was from Tokyo standing all the way to Akita.

3 Way:
Option 1:
Buy Grand Class Seat. You got 60% chances can get seat reserved during Golden Week. JR Rail Pass + 28,980 Yen
I called it Money Power.....Batman...''I am Rich''
Option 2:
From Ichinoseki Iwate take Local train heading to Sendai City. From Sendai take Night Express Bus to Tokyo. Expecting heavy traffic jam on highway but you still heading to Tokyo and all your plan won't effected. Just need to do some minor adjustment.
Option 3:
Booked the flight ticket now.
Sendai to Tokyo. One way ticket around Rm400-500. Of lucky, early bird ticket around 300-400
Until today some counter staff still practice this. In December during long weekend ( Tokyo to Nagoya) , the machine shows still have reserved seats but the new updated machine (load shinkansen ticket into Suica) can't accept foreign credit card, so I have to go and buy at counter. The staff told me all reserved seats are full throughout the day and only available after 6pm