This website might help you make up your mind - it lists a vast number of trains, showing if they have green cars and including many pictures of the train interior.

This page is the list of Shinkansen on the site, and the trains without the green cloverleaf symbol don't have green cars. On those that do, you can compare the seat size in the picutres. For example, on a non-Nozomi Tokaido Shinkansen, these are the normal seats and these are the green car seats.

You'll need to know the type of train that run on the routes you're planning on using, though.