Some sites have many questions with bounties on them. I'd rather not go through the ones that already have an accepted answer. Is there a way to sort questions to see only the ones that have a bounty and no accepted answer?
You can simply use search for that. Search for bountied questions by searching for unlocked posts with a notice, adding
hasaccepted:no to the search:
The search results include a few protected posts (link this one) but the majority of these are bountied questions.
Also see the advanced search help.