How do I cut lengths of wooden beading lengthwise?
The piece you have may have started life as the piece you just purchased and was ripped down to its current width.

The best tool for this is a table saw. It would be very difficult to do it any other way. many home improvement stores will cut down wood to custom lengths and widths. They may be able to cut it down for you in the store if you do not have access to a table saw somewhere else.