While developing my main game, where have a simple and old (dusted) sample on my signature, I has many problems dealing with DT.
So, what I had as problem, with my full custom movement system, was the player going into walls, because the frame rate update can variable and the system check the collision on each frame update, but if your sprite is in the eminence of a collision, and the frame rate drop down fast, the sprite can jump a large space in a short time, and when the system check it up, the sprite will be inside a wall...
After things like this, I started to use the platform behavior and work around all the other situations by attaching invisible sprites over the player, where I did custom movement system for each sprite, selecting them to be the main movement system as desired.
Another situation where the DT make my brain burn was while doing a simple car race. I did all the rotation of the AI using the DT as a multiplier, like you did, and I got low frame rates while playing on my smartphone (because of the physics), so, the AI started to miss the direction and go through turns, was hilarious, but serious sufficient to force myself to abandon the DT system.
Now, I just make it without DT and the system update right, but when the frame rate come down, the screen will show the AI doing a teleportation..