2000 Nissan Altima What's causing my car's loss of power?

I just had the catalytic converter in my Altima. Since I got it back, it has had some engine-loss-of-power-problems. I knew it had a muffler problem before the cat converter went but it apparently it now needs a new muffler assembly because the 'front pipe' broke. This is what my mechanic called it - you can see that the pipe that comes out of the muffler on the side facing the underside of the car (vs the outer, tailpipe end) is disconnected (looks like rusted out) from the big pipe that runs under the car. Anyway, my car runs fine for the first 5-10 minutes, then it starts sputtering and I can't seem to get it to 'go' - engaging the gas and clutch results in loud revving, the car barely going (or not going at all) and a terribly loud sound coming out. A couple of times I've had to just park and leave it, then come back in get it later (not far from my house, as I don't take it far). When I start it later and bring it home, it has been fine because again, it seems fine for the first 5-10 minutes.

My question is first, whether replacing the muffler assembly might fix this problem and second, if not, what could be causing the problem. I'm unemployed, don't have much money and really need to know the extent of repairs that are needed/how much it's gonna cost me, so I can determine whether I should just try and get a new car.

Any advice is appreciated - thanks for your help.