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I have a 2001 Buick Century. I'll step on the gas pedal, and sometimes it accelerates as normal, but normally it will lurch. Once in a while it will be extreme, but always noticable. This only started about a week, to a week and a half ago, and I've had the car for 3 years, so it's not likely user error or anything. Also, in the past week and a half, it has just shut off while moving a handful of times, and while I was on a quite road with my windows open, I could hear it maybe backfiring? as I tried to accelerate but only got lurching.

It reminds me of a manual car when the driver is not experienced with a manual transmission (i.e. lurching, & dying while moving), but it is an automatic. When I was in HS, my mom had a buick with a bad trans, so I was thinking maybe the trans is going out, but the lurching was different, and progressive. Mine is not progressive, but all of a sudden bad. Could this still be the transmission? What about the backfiring noice? Could that be the timing is off? Maybe the carburator needs to be adjusted? Would either of those get off what they should be, suddenly?

NOT getting enough fuel.

Several things can cause that.

Wet ignition, , but it would not stay wet for a week,

Water in the tank is possible, and also easy and very cheap to take care of.
The simplest fix is to add a bottle (they usually come in 12 ounce size) of isopropyl alcohol to the tank.The alcohol will mix with the water and let it burn away. The engine will not totally clear, but running will improve almost immediately.

Alcohol did not work?

The problem then moves to fuel supply, as you may not have sufficient fuel reaching the engine. Most likely cause is either a blockage in the line or a poor pump.
Again look for a cheap and easy solution first, try to replace the fuel filter.

After that you have to check fuel pressure and will need a gauge for that.

Could also be a mass air flow sensor (maf), or o2 sensor. The mass air flow tells the engine how much fuel to mix with the amout of air coming in. When the maf sensor goes bad, it wont usually up and quit, it will give to high or to low of readings. If it is too high of a reading, your car will pump too much fuel into the engine causing you to run rich. That will create a fresh gas smell like at a gas station. If the reading if too low, it will "lurch" or hesitate because there is not enough fuel coming into the engine, this also causes a screwy idle. Which when there is a low reading, it can sometimes die. Also, by th way, that car doesn't have a carb, it is fuel injected.

sounds like fuel. it probably has the 3.8 v6 in it so change the fuel filter and clean out the throttle body. the backfiring could be that the fuel and air mixture is not right and if it,s not getting enough gas it can cause it to backfire. also check the air filter too.