2003 Buick Century car stutters at low RPM when warm

Do you have a "check engine" light? If yes, you need to pull the fault code(s) out of memory. This will help diagnoss the problem.
Does it run ok above an idle going down the road?
My first thought is a gross vacuum leak somewhere. You may want to check the lines for disconnects or leaks.

It doesn't sound to me like ignition (coil). If it were, it should stumble under load. I'm still leaning towards vacuum leak. We can try an old "indian trick" here. Take a can of O2 sensor approved carb cleaner and spray around the intake area while the engine is at a warm idle. If you hit the area of a vacuum leak the rpms will increase and the motor will smoothen out.

I stand corrected. If your idle and rev-up is smooth, it's not a vacuum leak. Let me do a little research and I'll get back to you, Jim

This bugger is keeping me awake at night.
There are 4 things in the fuel system that could cause this problem. #'S 2-3 & 4 are supposed to be monitored by the computer and "should" trip a check engine light if they fault. I'll list them by "order of suspicion".
1- fuel pressure regulator (checking this comes later)
2- mass airflow sensor (MAF)
3- throttle position sensor (TPS)
4- crankshaft position sensor (CPS)
Suspects in the electrical system
1- an ignition coil
2- crankshaft position sensor (controls both injector pulse and spark)
3- Computer itself (ECM or PCM) same thing.
Let's unplug the vacuum line to the fuel pressure regulator and plug it. Also, looking for raw fuel in the vac line or at the regulator port. Take the car for a little spin, I wish I could ride along, and see how it runs. I'm not a parts replacer like, "change this part, oh well, that didn't work, let's change this." It cost too much and I like to be reasonably sure that what I recommend is going to solve the problem. Worse coming to worse, you may have to take it in and have it "hooked up" to a diag. Unit that can test the range position of all the sensors and the ign.
I printed a set of computer engine diagrams for your car. If you have acess to a fax machine, I can send them to you. They are more detailed than you get in most books. Let me know on the regulator test, Jim.

Fax was sent today.

If going down the road, normal driving, and you have lost the hesitation, I would say yes.

The only way you can check a CPS unit is to have it hooked up to a diagnostic unit. They can check the coils at the same time.