Upvote:1
You are right that ESTA is only for those entering under the Visa Waiver Program. Since Bulgaria is not part of the Visa Waiver Program, you will need a visa to transit in the US (unless you happen to have another citizenship or documents you're not telling us about). You can apply for a transit visa for this purpose.
If you are applying for a visa in Bulgaria, here is the local application information from the US Embassy in Sofia.
As an American, I'll say sorry. I know it's inconvenient and annoying.