@wizard After 14 years, I am surprised you find anything about your Basenji 'obnoxious' ! Surely patience with this old doggie is more than deserved after so many years. She might well be telling you it is her time - my Basenjis have almost always told me - or she might be warning you of the impending time.
If you find her whining obnoxious, could it be that your behaviour towards her has changed ? Are you being abrupt, less affectionate, changed in any way which could be confusing her ?
Time to start analysing your own attitudes, perhaps ?