Before you do a number on the starter, get a voltmeter somewhere. Ditigal Multimeters are for sale everywhere cheap. Put it on the 10volt dc scale. Put one lead on the starter input power terminal...red wire, and the other on the battery - connector. With your new fully charged battery, and voltmeter connected, put the ignition switch in the START position and read the voltage as you do. If less than 10v your problem could be dirty terminals between your battery and starter. Clean and bright and tight is the slogan and that's inside where the current flows (take them apart to clean properly not just on the outside.
That won't cost you anything but a little time and needs to be done periodically anyway. Then try again. If voltage is still low then go after your starter as directed. Nothing special about an outboard starter. If you have any auto engine manual lying around find the starter section and read up on how to check and clean a starter.