Angela
Read my section in the Segmented Pen article on the front page for starters.
On a bandsaw use a 1/8" blade. To help get a smooth cut, round over the back of the blade with a stone while the saw is running. I made a few bandsaw boxes this way and you can get a decent cut.
Use double stick tape to hold the two blanks together.
Here is the tricky part... cutting something this narrow... I would build a U shaped jig to cradle the two blanks to help them sit on the table safely, Mark you design and cut away.
If you do not get a nice cut, fold a piece of 100 grit sandpaper in half so the grit is the outside. Sandwich the paper between the two sections of wood. The idea is to sand both parts at the same time, so while holding everything together, and applying enough pressure, pull the sandpaper through. Repeat until you get a perfect glue edge. It really takes less time to do this then it takes to type it out, it's that simple.
Let me know if you need any more help.