I'm unable to recreate the problem.
I've checked the the widget , largecalendar shortcode, events shortcode, etc and are all doing what they are supposed to in my system. They either have the month nav off by default, or I can switch it off using the parameter:
Are you using latest versions of the software ?
Try [events listtype=4 show_month_nav=0] or
[events listtype=4 show_month_nav=0] in a page and check what that does for you. If that is right, the widget should be right too.
Check also in default theme, in case the theme is interfering somehow. If that is the case, let me know.
Else The only thing I can think of is perhaps try uninstalling, ensure the options are deleted and then re-install from scratch.
As you can see here, I don't have the problem on the demo site: