The hand plane is an excellent choice for flattening a board of the dimensions you described. It sounds to me like you may be having difficulty with where you are applying pressure to the plane as you're going across the board. This is a common problem for people who are new to hand planes. Basically, when starting the plane, the pressure should be on the front of the plane. As the plane fully engages on the board the pressure goes across the entire plane. As the plane is exiting the board the pressure should be on the back of the plane. If you don't follow this pressure sequence, you will likely plane and arc onto the face of the board. Unfortunately, if the board being flattened is already convex, the proper hand pressure will follow the convex surface and never flatten the board. See the second video for dealing with that problem.
Since you have a 13-inch planer, you may find it easier to flatten the concave side of the board. The concave side is much easier to deal with than the convex side. After you have successfully flattened the concave side, you can put your board through your 13-inch planer because you now have a reference side. I recommend finish planing the side that went through the planer or the surface will show planer marks after the finish is applied.
I attached two YouTube videos that you may find helpful.
https://youtu.be/D8K9dXuAvw0 around the 15-minute mark is a discussion on hand pressure.
If you must flatten the convex side first, the video below discusses the process at about the 16-minute mark.
https://youtu.be/2_96gNMMc_g
If you want to discuss this further, call me. My contact information is in the Member Directory.
Jim Russell
PS: Nice job on the shelf. It looks great.