I can see why someone would think there was Terrier, but I'm not so sure about Basenji. (The only way to know for sure is to do a DNA test.) Both of your dogs look very similar. What breed, or mix, is the older one?
Where did Nala come from? Can she stay there while you're out of town?
I did not have a good experience with Rover.com. Don't do it.
Can you find and make friends with another basenji owner in your area where you can take turns pet sitting? Maybe you can find local basenji owners on Meetup or Facebook?
My heart would break to leave her at a boarding facility or a vet although she'd probably be fine. I like the idea of a pet sitter staying at your home ONLY IF you can really train the pet sitter on several visits before leaving in August (not far away!!!)
Hope you guys don't get tired of looking at pics of possible Basenji mixes! This is Rio and I have tried to narrow his breed type to Basenji - Toy Fox/Rat Terrier - Chi - and Shephard. I keep seeing more Basenji than anything else but tell me otherwise! Thanks for your input!!!