Can you not afford to purchase another dog, or to care for another dog? If it's the initial purchase price, I will keep a look out for a second basenji on CL or petfinder.
Some dogs simply need a brother/sister. When Lola was an only dog (you can search on here for "Halo" posts from 2010), her family was at wit's end with her. There was even an incident when their neighbors called 911, and when the family got home, they were greeted by the police, who insisted on entering the house and seeing the creature that was making all of these horrible high pitched squeals!
Fast forward to Lola's life with us, and I had to move into an extended stay hotel for 3 months, I figured I could take Lola with me and she would be just fine, as we had never heard Lola make one peep! WRONG. For a dog that doesn't bark, yodel, or roo, Lola can cry to a noise beyond belief! Since I couldn't leave her at the hotel, I brought her to PetSmart doggy daycare, thinking that would cure her anxiety. After all, she would be surrounded by dogs, right? WRONG. Lola would cry so loudly that they "upgraded" her to a soundproof suite…. I thought it was because they were pampering her, but later realized it was because you could hear her cries within the store itself.
Over the weekends, Lola and I would go to the dog park and outings, and she was just fine, as she was always with me. However, one time when I stopped to get gas, I had to go inside to pay. When I came back outside, there was a gathering around my car by concerned citizens~ what was wrong with the puppy? Was she hurt? (This was when i realized how LOUD she can be!)
The way this was "solved"- in an instant- was to give her a sister. Once we adopted Zeba, Lola's anxiety was gone. While the two of them still went to doggy daycare together, Lola would happily and quietly share the kennel with her sister. She was truly a different girl.