What I observed was that when I was using an account with dba permissions, I wasn't getting insight at all, regardless of whether it was the first line on a new tab or in the middle of a script. As I stated in my original post, when I used CTRL-T, the popup was completely blank and hitting refresh had no effect. I tried closing toad and re-opening it, and the behavior continued. I tried refreshing insight several times, with no change. I then logged in with a non-dba account. With that account, insight worked immediately as expected (i.e. popping up as I typed and the popup was populated appropriately). I tried going into my dba account several times through the course of the day and it continued not to function with that account. When I opened Toad this morning and logged in with my dba account, insight started working properly.
So, I don't know. It's apparently just a weird glitch that happened that first day. (By the way, yesterday was the first time I'd opened toad after installing that version of the software, if that is of any relevance.) It seems to be working as expected now.