I've got a Garmin Nuvi that's at least that old, if not older.. The battery holds a charge for about 10 minutes when 'charged' to full...
I don't have those other symptoms you have while plugged in, so not sure if just changing the battery will resolve all of that..
I did get the lifetime map updates from day one, and it still updates to this day.
As far as I know, even if I get a new Garmin, the same lifetime map update will work with it too, but I don't feel the need to get a new one at this point..
It works great, has options for different voices and icons and I like that! I have been using the "Dr Nightmare" voice forever and it's just dang funny..
Anyway, I know my phone map apps are better, but I still pull out the Garmin on my camping trips!
Good luck!
Mitch
2013 F150 XLT 4x4 SuperCab Max Tow Egoboost 3.73 gears #7700 GVWR #1920 payload. 2019 Rockwood Mini Lite 2511S.