Update: i reset my settings, and now everything works...well, as well as it can under iPadOS13. Thanks to Feenician and melancholy! So...if anyone has my same problem, the solution is: settings -> general -> reset -> reset all settings.
Great news that it's working. The new way of doing things does feel like a big regression to me but I did find I got used to them fairly quickly, even if I have not grown to like them.
The Spotlight one seems like a flat out bug (there is no useful byproduct of not allowing you to drag out and drop without hitting cancel) and for the home screen I'm guessing they changed it because having two modes of drag and drop (wiggle mode and non wiggle mode) did not sit well with some. These are just speculation on my part though.