I need to upgrade to JEvents 3.2.19, but I don't have the automatic Joomla updater. Do I need to uninstall the current version before upgrading, or will the upgrade just overwrite the existing files?
No - don't uninstall, Joomla will remove your categories data if you do that. Simply install the new version on top of the old one.
An up to date site backup is always a good idea before an upgrade