If the iPad Mini 3 has iOS 11, that would be the a 3-year difference in age between iOS version and hardware version, so that could introduce some speed issues.
But some of what you're describing here other explanations:
After it does eventually move, I can't move it again for another second
The app now ignores taps that come within a second of a drag. This is because many people accidentally touch the screen with a stray finger while dragging, triggering unwanted page jumps.
When I tap my screen to jump to the next page there is also a delay (about 300 ms I'm guessing)
This sounds like the delay needed to determine whether you are single-tapping or double-tapping (the app has to wait to see if a second tap arrives before acting on the first tap). That hasn't changed in years as far as I know. Anyway, do you really need to change pages with sub-second precision?
When I just try and move my lyrics with my finger, it does not respond right away and sometimes not at all
A video would help troubleshoot that.
Hitting save after editing automation takes a long time now.
A help desk ticket would help troubleshoot that.
BTW, it's likely that jumping to a marker is slower than jumping to pages as before, but if you don't have markers, the speed would be the same as before. In other words, adding the new functionality shoudn't change the app speed for someone who isn't using the new functionality.