Perhaps you could try training an incompatible behavior. You need to observe him closely so that you can anticipate what triggers the biting (it sounds to me like he is trying to play with you and doesn't know it's inappropriate). When you think he is about to bite, cue the behavior you want to replace it. Since he knows sit, that might work, but may I suggest something involving his mouth?
Rather than hand him a toy as a substitute for your hand or leg, train him to "pick it up". That way he gets to put something in his mouth himself, and be praised for it. (does he like food rewards? Find something particularly salient when you start the training so he will want to work for that instead of biting you, which is obviously quite salient too!)
Incidentally, "pick it up" is one of the more useful things you can train a dog. It also leads you to that which most people don't expect…...a retrieving Basenji. This is easiest clicker trained, and it is also a lot of fun. Here's how to proceed. Have an object on the floor. If can be anything that is easy for him to put in his mouth......but not something too small. Clicker train any approach to the object, gradually requiring closer. Soon he will get his mouth near the object. Eventually (often sooner than you think) he will sniff the object, touch it, and finally the mouth will open and he will pick it up, at which point, jackpot!
Once you have a consistent picking up of the object, you can put it on cue (say "pick it up" as he does it) Once he is solid on the command you can proceed to having him give you the object. You can move on to a retrieve by placing the object further away, eventually out of the room, etc.
The real gem in all this is it leads to identification of objects and retrieving the thing you want by name, but that is a subject for another time. In the short run, you want to replace the behaviour you don't want by one that will hopefully be pleasing to you and him, and distract him from biting. Make sure you have a suitable toy handy for him to pick up, and when signs of biting appear, try to redirect his energy to picking up the object, giving you a chance to praise that action while dissuading the other. Worth a try!