Shinkansen on December 29 and January 4?

I have a trip to Japan planned from December 26th to January 11th, I will be traveling from Tokyo to Kyoto on December 29th and from Kyoto to Hiroshima on January 4th, but have now read that these dates are extremely busy for travel in Japan. I have also already bought a two week JR Pass.

Am I going to run into problems to find a train to travel in? If so, what could I do? From what I understand, I can't make train reservations with my JR Pass until I exchange my order for the actual pass, I just want to know what to do in order for these days not to be a complete disaster.

The rush will already start on December 28 this year. So, I don't think that the 29th will be as busy as usual. I suspect that if you reserve on December 26, there will be some seats left on trains in the afternoon or evening of the 29th, but maybe not in the morning.

I have a feeling that Kyoto to Hiroshima on January 4 should be no problem if you make reservations on the 28th. The retrun rush will be distributed over January 4-5 (which is a day longer than usual), and the section from Kyoto to Hiroshima isn't as crazy as the section from Osaka to Tokyo would be.

Am I going to run into problems to find a train to travel in? If so, what could I do?

In case all seats are booked out, you can still ride in non-reserved cars. Go to the station early and line up on the platform for non-reserved cars. You are not allowed to use Nozomi and Mizuho trains with the Japan Rail pass, even if you stand.

And you always have the option to stand in any of the Shinkansen trains, or sit on your luggage in the area between car wagons.

I think you will have problems both days. For 2018 both the Saturday and Sunday before NY were fully booked (every single available seat on every shinkansen across both days) on the Tokyo/Osaka route and even reverting to non-reserved cars, you may have to stand unless you pick an unusual time. Certainly you would want to book/reserve your seats as early as possible.
For the trip to Hiroshima, you get stuck with JRWest and their limited capacity for JR pass users. The Sakura trains sell out and many people use the non-reserved cars. Even on ordinary days it can be crowded. You might be able to use the Kodama service even if it takes longer. Again, try to reserve seats as soon as possible after arriving in the country.