It's not easy to promote audiobooks, at least those done through Audible/ACX. You can't run a sale because you have no control over the price.
One thing you can do to help get reviews is to request free promo codes from ACX.
You will almost always be able to get 25 for a new book, and sometimes I've gotten additional codes down the road.
You can then submit your book to Audiobook Boom.
For $10, Audiobook Boom (if it accepts your book), sends a listing about your audiobook to its list of almost 5,000 subscribers.
How Audiobook Boom Works
Those who are interested in listening to and reviewing your audiobook request it. You are not required to send any of them a promo code, though.
Audiobook Boom sends you each requester's name, email address, and review profile on Audible (or sometimes Goodreads). You then look at the profile and decide if you want to give that person a promo code to use to download your audiobook for free.
I just did this for the fourth (and last) book in my Awakening supernatural thriller series, The Illumination.
When I checked profiles, I looked for people who either (a) had reviewed a lot of books (many had reviewed hundreds) or (b) had reviewed at least 6 or 7 audiobooks that fell within the supernatural thriller, suspense, horror, or occult genres within the last year or so.
One person had reviewed 5 audiobooks but judging from the bare man-chests on 4 out of 5 of the covers, they all had a strong romance component.
My Awakening series has an occasional sub-plot involving a romantic relationship between characters, but it's minimal and, for one couple, occurs entirely off-screen. So I didn't send that person a code.
If a person uses the code to download your book, you do get some credit on your Audible sales dashboard.
It's hard to say exactly what that translates to in dollars, but I'm pretty sure every time I've used Audiobook Boom it has paid for itself.
There are possible downsides.
(1) There's no guarantee that your audiobook will be requested. I've always had at least 10-20 people request each book in my series. But even if no one does, you've only lost $10, so I think it's worth a shot.
(2) Not all requesters actually leave reviews.
That being said, Audiobook Boom does ask that you report if people don't leave reviews, and that eventually may take them off the subscriber list.
(3) Audible has changed its practices. You used to be able to use the promo code yourself to send your book as a gift to the requester. Now instead you send the person the code with instructions on how to download your book. There is nothing, though, to stop the person from using the code for a different book entirely.
(4) Finally, there is the obvious possible drawback–people are not obligated to leave a positive review.
For the most part, though, if your blurb and cover accurately signal the genre and your audiobook is of reasonable quality, most people will be fair. At worst, if they don't like something, they'll say why, and you may learn something for your next book.
Let me know if you try it out.
(For more on creating audiobooks see 3 Ways To Create And Distribute Your Audiobook and The Cost To Create An Audiobook Edition Of Your Book.)
P.S. If, like me, you are posting an audiobook that is not the first in the series, you may want to make sure you have codes for previous books available as well. I'd rather give someone an extra code to try the earlier book first than to have them try to listen from the middle and not understand what's happening.