Tokyo to Osaka shinkansen reservation?

I am used to just buying a train ticket whenever I want to use the shinkansen, but I have never taken the shinkansen this route before, and especailly with the new luggage rules I would appreciate some insight.

I prefer to leave my options open so that's why as far as possible I generally only buy tickets when I am actually about to travel, since I do not know if anything will delay me from getting to the station. One of the main advantages to me for shinkansen vs flying is I don't have to worry about making it by a certain time, the trains run frequently enough and my schedule is pretty loose so taking the next train half an hour or an hour later is fine.

However that all may have changed due to the need to buy certain class of tickets when hauling luggage.
We will be travelling from Tokyo to Osaka on a Sunday, and returning from Okayama to Tokyo on a Friday. Should I be concerned at all that they will entirely run out of tickets with luggage included?

The luggage rules are only applicable for very large luggage. More normal luggage can be put overhead.
If you take the Shinkansen from Tokyo (as opposed to Shinagawa) you can just line up a bit early for the unreserved and get a seat in the last row and use the space. I have done that multiple times when lugging around my road bike which is much bigger than any big suitcase. And no problem.
I am also one of those unreserved freaks. No one seems to understand me, but I find unreserved between Tokyo and Osaka just more convenient. No need to wait for a train, just hop on and off you go.

Should I be concerned at all that they will entirely run out of tickets with luggage included?
If you are looking for maximum flexibility and booking at the last minute - yes. Even before the tourists returned to Japan in large numbers at the end of 2022 the luggage storage seats on the Nozomi trains would get booked out (let alone the trains for those using rail passes).
Better approach is to make sure your luggage is under the 160cm limit that has been in place for the past three years so you do not need to reserve those specific seats.

why don't you send your luggage by ‘î‹}•Ö ?