I have fostered many basenjis, ~200, from many different backgrounds and have b-sat intact males and females. I regularly have more than ten basenjis running together. Some seem to prefer pals of the same sex. I've had to be more careful with some males together, but the same applies to females. The ones more challenging were larger, so my experience has been that more males were a little more difficult to integrate than females since the males tend to be larger. Perhaps their slightly larger size made it easier for them to intimidate prior owners and set up a behavior pattern that was difficult to change.
To minimize the potential for issues, they are not left together unsupervised until it is quite apparent that they know our rules - no one gets to be top dog. No fighting allowed. Even when they are playing, I end the fun before it gets out of hand. I have seen many times when a couple started to get too rough. They do not forget quickly, so I try to make sure no one has a bad experience particularly prior to having lots of good interactions.
Note that all of this may go out the window if one gets sick. I have had fosters who were fine in adoptive homes for a year, when suddenly attacks started. As one example, the attackee ended up having lymphoma and had passed within half a year of the first attack. One really just has to constantly be vigilant because the game can change at any moment. That's part of the fun with basenjis.
Like others, I am not a huge fan of dog parks and have had an unpleasant experience with an unpleasant GSD owner who felt he should kick all three of my dogs who were trying to get his on-leash GSD to play in the off-leash park. I still value dog socialization, though. There are other ways. You might visit a local dog club to meet owners who might agree to go on walks with you and your b in neutral territory. You can walk in the area of a dog park and again, perhaps meet an owner who is willing to go for walks. After making friends with a buddy outside of the park, meeting at the park when there are few other dogs around for off leash play may make the park intro pretty simple and less intimidating.
Lots of good input already. My thoughts are that he needs confidence building. Very short (5 minute) training sessions to work on commands like sit, stay, leave it, look at me, come, touch, etc., a few times a day, definitely with every meal, should help. Once he does a command reliably with you, have the kids work on the same command. This gives everyone a clear reason to praise him while building communication. Children can bring a lot of energy into a home which can be pretty intimidating when one is a little guy recently separated from mom and littermates. Having them participate in his training in a supervised manner will help them to appear less scary.
I prefer to leave soft rock or sitcom TV on almost all of the time. It helps to avoid a lonely silence while taking the attention away from complete focus on whatever I'm doing. It also masks the sounds of school kids walking by, squirrels running through the trees, other dogs, etc. If I don't leave on something to break the silence, my guys hang around on edge, waiting for the slightest noise to spur them all to jump up and run to the window with hackles raised.
I have little squeaky 'hearts' that I kept from destroyed toys that I can take out with me. The squeaky noise gets them to run to me (some are blow-through and can be very compact).
Have you tried real meat as treats? One of my formerly anorexic basenjis liked desiccated chicken (I never understood the appeal of 'cardboard' like meat to him). Is there anything special that he really likes (attention or toy, etc.) that might be considered 'a treat' to him? To reinforce the recall, a treat can be given every time the toy squeaker is used at home in a controlled environment so that it may still work when you are out at the park. Just save it for "really need the recall" times, though squeaking it when you are back at your car and can safely give him a treat would be a good idea.
As many have indicated, stop the frequent bathing.
Note I use gel caps for Omega 3/fish oil. It helps to reduce the potential for the fish oil to degrade by being exposed to air.
You may also want to have thyroid function tested. Hypothyroidism may be accompanied by a course, dry coat. How old is he?
A light coat or jacket, light enough for your inside temperatures, may polish his coat and reduce the amount of contact hives you experience while cuddling.
I have tried it for a b girl with corrected ectopic ureters. While it might help a little, it was inconsistent. She will sometimes have soft stool regardless of medication. Both her stool quality and urine leakage amount seem linked to her state of mind. When she is more excited or hot, she will leak more. Too high of a dose of Proin also caused her to leak more. Her optimum daily total was less than that normally prescribed and was most effective if equally spread over at least three doses equally spread throughout the day. Her bladder leakage seems a little more like bladder spasms (not 'accidents'), so a different medication for spasms did work a little better, but I doubt it would work well if she were guzzling like your b is.
I have for now found that a thong-like diaper with a little pad works best. Her mind doesn't stay calm and consistent, so medications don't work consistently.
Is one more potty break an hour after guzzling an option?
I have used Embark DNA tests. It does come back with basenji as expected. The health content has also been very helpful. When I was testing, the reviews on Wisdom were not as good. Perhaps you might want to try another test someday (it is on sale now, $145 includes health - there has been a sale for even less, but not sure if they'll do that again).
She really has great eyeliner!! Beautiful!
I had a foster who tested via the Thyroid5 panel at hemopet to have hyperthyroidism which may indicate a cancer in some. He was tested right before arriving at his foster home and was quite active when he first arrived, running laps around the yard. We waited on further testing, giving him a couple of months to chill by the pool. He was pretty stressed out and confined in his previous home, so chill time really was needed. Two months later, another complete panel returned perfect mid-range normal results. No medication was ever used.