How can I avoid this?
Use a router to cut stopped dados instead. If you're limited to using a table saw, and that saw doesn't happen to have a router table built into it, it's pretty much unavoidable.
Is there a trick?
Only if you consider using a tool other than a table saw tricky.
A different kind of corner joint?
Sure, that'll work. I thought you ...
So what do people do? Do I buy an 8" dado set for the right arbor size? Or do I get an arbor adapter to use with a 10" set?
You do the former. 10” saws use 6” or 8” dado sets. 10” dado sets are for larger saws. Avoid so-called “wobble” cutters — those won’t cut a flat bottom and will just frustrate you. 6” dado stacks are OK if you’re on a budget ...
For general advice on cutting dados the Related Q&A is where you should start. The core question can be directly answered using previous Q&A, but I've collected that here for easy access.
My advice would be to use smaller blades and spacers for the arbour you have to keep it simple and reduce the overall moving mass. It is generally fine to run ...