Hello Ryan
When doing segmenting you will find there are many ways to accomplish what you want to do. You need to find a way that works for you.
It the type of segmenting you are currently doing it is a simple design and is called stacking. There are 2 ways to make those type blanks. Cut all the pieces and glue together and drill for the tube afterwards. Or you can drill the individual pieces ahead of time and now stack them on the tube, gluing as you go. Doing it that way you will not get the bit to wander because the pieces are not very long. The longer the piece is and the more different woods you stack (the grain patterns are different with each spiecies of wood) the more the bit wants to wander.
There is no measuring of the tubes. Just take the woods you want to use and stack them in the order you want. Now lay the tube next to the stack and center the tube on the stack so that you have the same equal pieces on both sides of center of the tube. Simple. You now will probably have to cut off slivers from both ends of the stack. Just allow a little on each end for trueing the blank latter. Just keep the tube in the center of the stack.
Here is my jig I use for cutting slices. very simple jig that I use stop blocks to keep the size of the pieces equal. I use clamps to keep my hands away from the blade when working with small pieces. You can do the same type jig on a bandsaw too.
Also here is an example of stack cutting and assembling. I cut the pieces before I placed them on the tube. I then used epoxy glue and glued all piece and also glued the tube in together. Works well for that type segmenting.