Is there a convenient way to search for answers of mine that have been awarded bounties? I know that I can look through the "reputation" tab of my user profile, but this is very tedious, especially if I'm looking for a bounty from long ago.
On the Activity tab of your user page, click on Bounties (in the string of buttons below the reputation graph and Newest/Next Badge information). That string starts with Summary and ends with Votes, and Bounties falls just past the halfway mark.
In the Bounties mode you have three submodes (buttons to the right side of the desktop page), Active, Offered, and Earned. Click on Earned if you want to see the bounties your Answers have earned.
Here is a SEDE query: Answers by a specific user which received bounties. By changing the uid parameter you can check which user are you looking at.
Of course, you can refine the query using further criteria.
Keep in mind that the data in SEDE are only updated once a week.