I don't know if this will be much help. I'm interested in the answer, too, because I'd like to visit Azerbaijan myself.

There's a trip report from late last year on Turkmenistan that mentions STANTOURS. If you read the thread, it appears that this outfit has a good reputation, perhaps not the cheapest.

They don't mention the e-visa option on their website, but they do handle Azerbaijani LOIs. If I understand it correctly, you can't get an e-visa without the LOI anyway (and yes, you get stuck paying for it. Not the only country that does this). You might consider emailing STANTOURS to see if they have any more information.

In the past I've read that the e-visas were never intended for independent travellers, only for groups, but that may or may not be true.

There was a travel fair in LA that just ended. A few Azerbaijani travel companies were represented there. You might also try contacting one of them if you keep hitting a dead end with the embassy itself. Presumably if they had it together enough to show up for an LA travel industry show, they're more likely to be able to handle international inquiries.

Good luck! I'm going to be facing the same process (whatever it is) myself eventually.