I've been lurking on these forums for awhile and have use the information here to my advantage, so I figured I'd return the favor and possibly help someone else out.
My wife's I-485 application has been pending for nearly 2 years now (USCIS lost the application when we moved and we had to go through our Senator to find a solution). We didn't apply for AP originally because we expected a relatively quick adjustment timeline and didn't have any immediate travel needs. This summer we decided we were sick of waiting and wanted to travel. So we sent in the AP application in July and received the document in about 3 weeks (super fast!).
Similar to most of the experiences on here, we were nervous to use the AP but decided to book a cruise with some friends. It took us a few extra minutes to board at the cruise terminal (Miami) because the cruise employee had never seen the AP document before. She went and spoke with her manager and cleared us to board. When we returned, we went through the immigration line. The officer checked my passport and we gave him my wife's Mexican passport and AP. He told us we'd need to go to the secondary screening room so they could verify that everything was okay. We went to the room and sat there for about 10 minutes while an officer typed a bunch of info into the computer. He seemed kind of inexperienced with the AP document, but was nice about it. He asked us if we had been told when we were going to receive the green card and we told him we weren't sure, but hopefully soon. He returned the AP (both copies) without any stamps. I asked him if he needed to take a copy and stamp them and he said, "Nope, you're all set to go." So we grabbed our things and left.
It was a pretty relaxed entry process, and I'd imagine that airport entries are a bit more strict. But my advice to anyone looking to travel on AP is to go for it. It was quick and painless.
I'm curious, has anyone else had a similar experience with the officer not stamping documents? He did take my wife's I-94 (she had left it inside the passport), saying she didn't need it anymore with the AP.