I'm guessing you've already confirmed that the cigarette lighter in your car actually works.
Return it. All the Garmin units should run off your AC car adapter. If your unit doesn't, it's defective. It happens.
But the main point is: shouldn't the GPS unit be able to run off the power of the AC car adapter, when plugged in? I can't imagine the answer being no, as that would be a mind-bogglingly bizarre omission.
You should see the following behavior on all modern Garmin units:
You should get 5-10 hours of battery life, depending on the model. The GPS charges whenever you have it plugged into the cigarette lighter.
When you have the unit plugged into the cigarette lighter, and turn on your car's ignition, the GPS turns on at the same time. When you turn off your car ignition, the GPS asks you whether you want the unit to stay on or power off; it counts down from 30 seconds and turns off automatically if you don't touch the on-screen button to stay on.
Well, I really went for it during boxing day. Not only picked up the new car stereo but nabbed a Garmin C330 GPS while I was at it (my first GPS).
If you bought the C330 locally, you may have overpaid. It's $325 at Amazon, and you can get the C340 with vocalized street names for $399, or the C530 without vocalized street names, but with a better screen and GPS chipset for the same price. The Garmin Nuvi 350 @ $460
gets you all of the above -- it's also what I recommend for a GPS under $500.
What's the benefit of vocalized street names? One announces "Proceed four miles on Glebe Road and then turn right onto Lee Highway" followed by "In point-two miles, turn right on Lee Highway" (varies by speed limit) and then "Turn right on Lee Highway" as you get very close. On the other unit, you just get "Proceed four miles.." followed by "In point-two miles, turn right" and then "Turn right." On both units, the street names are clearly labeled on the map but when making every turn, you don't typically look at the screen.
The announcement of street names helps in two ways. When you know the area, it suggests to you what route the GPS is going to take. When you don't know the area, the street name announcements give you a better idea of what to expect with the turn, because you know well in advance what street or highway exit you'll be turning on.
The Garmin C340, C550, Nuvi 350, Nuvi 360, and Nuvi 660 all announce street names. The Garmin C330 and C530 do not.