Jerking issues at complete stop

Hello fellow Aveo owners! I am having an issue that I have never experienced before with any car I've owned. I have a 2006 Aveo LS with roughly 80,000 miles. When the car gets up to operating temperature, it will drive just fine. When I come to a complete stop, there is a jerking feeling almost like it's trying to shift into gear. It will do this every 15-20 seconds at the stop and then sputters when I begin to accelerate. I have been putting it into neutral to avoid this, and it seems to help. Then when I get home and back the car up, it really stutters going in reverse. When the engine is cold, I never have this issue, but once it warms up, it is a regular thing. The check engine light has not come on and the transmission was flushed less than 6000 miles ago. This has only started to happen in the last 3-4 weeks and was wondering if anybody else has run into this issue before. Thanks in advance for any advice :)