So far things are working well, although I still have a few things to move over. But I found one thing I don't like about OneDrive. It seems to be tied in to both of my computers' Recycle Bins. When I delete a file (online or offline), the deleted file shows up in both Recycle Bins. This seems to be by default with no means of disabling it. There's nothing in the OneDrive settings to control this.
The setting is also pretty aggressive. For example, if I delete a file on Computer A or on Computer B, it will show up on both Recycle Bins as well as on the online's Recycle Bin. If I delete a file online, it then shows up in the online Recycle Bin along with both of my computers' Recycle Bins.
EDIT: I completely disabled the Recycle Bins on my computers. It seems that's the only fix.
EDIT AGAIN: Disabling the computers' Recycle Bins seems to be the best solution, at least for me. I have never used the Recycle Bin, so I won't miss it. The deleted file still shows up in the online's Recycle Bin, but I'm okay with that as it's simple enough to manage.