I did some more troubleshooting today. It seems my first description wasn't quite accurate. The RPMs weren't dropping, they were increasing slightly. Once I realized that, I found that I was able to trigger it by turning on the lights or fans. This leads me to believe that the change in RPM was caused by the alternator clutch locking. The increase in RPM was applying more torque to the transmission, which was giving that pull that I was feeling. This seems like the normal operation of an automatic transmission and I'm freaking out about nothing.
I was checking the trans fluid with the car warm but off, so I'll be sure to do it right in the future. Thank you guys for putting me on the right track.