Just a little extra advice on the charging of the Nuvi, you mentioned that when on the vehicle lead it still showed low battery, this can only be one of two things.
As Greg has said this can point to a faulty battery (in this case shorted) and so if this proves to be the case then you should return the unit under warranty and accept Gregs answer.
The other thing it can be is a faulty vehicle lead, you need to try the following:
1. Text your Cigarette lighter socket with another device or preferably try the unit in another car that has a known working socket
2. Check the fuse on the lead, this is in the cig plug on the lead, just unscrew the end and you will see a glass fuse inside
3. If still no luck then try connecting the Nuvi to your computer via a USB lead and see if it will power up in PC connect mode
If still no luck try resetting the unit by pressing the reset switch with a straightened out paper clip or holding the power switch on for 10 seconds.