I now you say you don't want to drug her the rest of her life, but look at in another way; how many people in your family or people you know have to take a daily medication. We may not want to but we have to because there is an imbalance in our bodies so we have to take the medications to help us balance out.
I have to take medications 2 times a day, not by choice really but if I don't I become very ill because I suffer from debilitating migraines, and the meds I take are preventives. If I don't take these drugs I seriously cannot function at all, I get migraines to the point the I literally loose vision and my other things, it's bad. But that's my point I know we don't all want to to take meds or give meds, be in that situation but if it will help your animal it's a better choice for them.
Crate habits - start all the way from square one again, limit the size of her crate; dogs will not like to potty in the same place they sleep this is true. Okay you are thinking but she does and then steps in it or maybe just goes in one corner of her crate and then stays smooshed into the other side of the crate.
I have a foster from BRAT and he's about 24wks now and when I got him he had little to no potty understanding for the crate. I had him in a medium size airline crate, and he decided to "finger paint" in his crate gag my husband and I had to carry him outside before letting him out LMAO after that I took a round laundry basket that I had and then a cardboard box and stuck those in the crate and it made so the crate was about a 1/3 the size. The pup literally had just enough room to squeeze himself turn around and lay down; he couldn't fully curl up in a basenji ball. But with that he did stop peeing and pooping in the crate. I left his crate like that for a week, then I tried taking everything out, nope he peed in it again, I stuck the laundry basket back in. Another week went by then I took the basket out, now we have gone 3 days with a dry crate when ever I have left.
I am like you and only leave for short periods also.
Try this starting this training method with her: http://www.basenjiforums.com/showthread.php?t=2055&highlight=NILIF basenjis are such stubborn little dogs LOL and they will test you at every chance
Here's another good one from Tanza (about just placing yourself as the pack leader): http://www.tanzabasenjis.net/welladjusted.html
Sometimes with rescues (even adults) you have to start all over like a clean slate a fresh puppy slate - basically pretend you are getting a puppy. And unfortunately it sometimes happens that you won't notice changes or behavioral difference until after they have "settled" in for a bit (sometimes months).