'07 2.2l LS is surging at idle and sounds like a lawn mower when accelerating. Advice?

I recently acquired a second-hand Cobalt. I was told that it has a leaking head gasket and I'd like to confirm with some Cobalt owners that is indeed the main problem.

The main issue is that when I start it up and leave it idling, it begins to surge. The longer it is left idling and surging, the harder the surges become and it violently shakes the car. I left it in that state for a few minutes and came back to it with so little power to the surges that it shut off in front of my eyes. I turned the key slightly and the engine started up again.

I checked the codes and there is a P0304 code for cylinder 4 misfire.

Lastly it just doesn't sound like a car at all, it sounds like a sit down lawnmower when I step on the gas.

I'm a bit broke at the moment so I want to want to be 110% sure that the head gasket is the problem before I order the parts. Is anyone available to help me confirm that?