There are two types of USB cables. One type supllies data and power and the other only supplies power. The data cables have extra wires and can confuse the nuvi into thinking it's connected to a computer. I have a whole milk crates full of USB cables, some of them will do what you describe, others work properly.
So you just need to find the correct kind of cable. You might try searching for USB power cables on Amazon. Unless there is something wrong with the Nuvi, that is most likely the problem. You could also go to a store like Best Buy, get a power adapter, take it out to the car and try it, then immediately exchange it if it doesn't work.
I got one at Radio Shack a number of years ago and it worked, but that won't be an option any more. Also got one from Staples, they were cheap.