My 2004 Pontiac Bonneville has this problem sometimes when I start it up in the rain, if I start it up in the morning. It will sputter while I am in drive or reverse, and driving below 40 MPH. When I hit 40, it goes away and then will come back when I slow down or stop. It usually goes away after 20-30 minutes of driving.
This never happens when it’s dry. My car runs smooth and normally when it’s dry. When I took it into the shop, they could not find a problem.
Find a different shop. It’s easy enough to spray water on parts of the engine to find this problem. Which probably is the ignition wires.
One of your coils is probably cracked and shorting out due to moisture. If you look under the hood in the dark, you may be able to see the arching from the coil.
I agree with @BillRussell And @knfenimore, especially since it’s moisture related. Coil(s) and/or ignition wires. If it’s coil(s) it may be internal in which case you won’t see arcing.
This looks like a problem in the high voltage part of the ignition system to me also. I’ll have to disagree with @BillRussell a little on the point of whether this is easy to diagnose or not by spraying water on the various parts. Often this method works, but not every time. I had a VW Rabbit years ago with what turned out to be a tiny crack in the ignition coil, and I could spray water all day long, from any angle I could physically get the hose to spray from, and it would never stall out. But drive it through a big puddle at over 25 mph, wham, the engine would stall out immediately.
Remember those packages you get at Christmas saying “Some Assembly Required”? In your case OP, some disassembly may be required to get to the bottom of this.