Most of the time, "but" is used to join two independent clauses into one compound sentence. In those instances, there are three obvious ways to eliminate the "but":
1. Split the compound sentence into two simple sentences.
(Before) “The Hall of Records in Wistica sends us information about future Avatars when they're born, but we’ll have to search through her office for the records.”
(After) “The Hall of Records in Wistica sends us information about future Avatars when they're born. We’ll have to search through her office for the records.”
2. Replace the "but" with a semicolon. (This may change the relationship of the sentences. Some people feel that semicolons don't belong in fiction; I say they can work if handled properly.)
(Before) She must have done it in other lives, but would she be able to do it in this one?
(After) She must have done it in other lives; would she be able to do it in this one?
3. Replace the "but" with a semicolon and a "however," with a comma after "however." (This approach keeps the meaning intact but still bothers the semicolonphobes. I'm sure I had an example of this; however, it would take too long to find it again.)
If you're willing to rewrite your sentences, you can come up with some other ways to remove the "but."
4. Rewrite one of the independent clauses. (In the example below, I moved a phrase from the second sentence to the first. I had some other examples where I made the second clause dependent on the first.)
(Before) Aunt Gabri joined them as well. Her color was too pale, but she watched Kron as if she shared Gwen’s suspicions.
(After) Aunt Gabri joined them despite her pale complexion. She watched Kron as if she shared Gwen’s suspicions.
5. Change the "but" to an em dash. This may require a rewrite.
(Before) She wished she could study the pot close-up, but she wasn’t sure if it was safe to do so.
(After) She wished she could study the pot close-up—if it was safe to do so.
Finally, you can remove the word (6) (if it's at the beginning of the sentence), one of the independent clauses (7), or the entire sentence (8). Here's an example of number 7:
(before) I understand, dear, but you keep driving and driving yourself without a break.
(After) You keep driving and driving yourself without a break.
I wound up removing between a third and a half of the marked "buts." Of course, I'm still sensitive to the word and keep finding more instances that no longer feel appropriate in context.
Are there any particular words that bother you when they're used too much?