Tim
I don't know if you have a table saw. If you do I would use that to cut the segments. You will get a much cleaner cut. Ideally there should not be any voids to fill while turning. The two faces that are being glued together should be perfectly flat and flush to each other. If you do not have a table saw you can achieve that with the band saw and disk sander, but it will take a lot more work with the disk sander. You should be able to hold the pieces together up to a bright light and not see any gaps or light coming through.
When I am gluing wood to wood I like to use Titebond II. It does not have a very long open time, although the open time is longer than CA. Initially I would glue the pieces together using a "rub joint". You take the two pieces and rub them together for a minute or so until you feel the glue is starting to set. Then double check the alignment and start with the next rub joint. Once all the pieces are glued together you can set it in a jig and clamp it. This technique and the use of Titebond will only work if your joints are flat and flush to each other. Titebond is not a void filling glue.
Hope this helps.