Typical Contax IIa & IIIa shutter problem at high speeds (1/1250 for sure, can happen at 1/500 or even 1/250 too) : the first curtain begins its journey with a tiny bit of hesitation due to either grime on its beveled gears or dirt and/or oxydation within the curtain shafts housings (the latter being nasty because the shafts housing are sealed and so, oxydation spots in there cannot be removed).
Sometimes a complete stripdown of the shutter box followed by ultrasonic cleaning of the gears and shafts (not the curtains themselves of course) and proper relube using Nye or Moebius oil typically applied with a syringe where it ought to be can fix it.
Sometimes not especially is the camera is a "colour dial" model, their shutter release mechanism being quite complex (and a bit funny). If you add to this the dumb design of how selftimer delays the shutter release you have something really interesting as for what should not be done when it comes to focal plane shutters design, but Zeiss Ikon liked to prove that they could manufacture the most complicated shutters that shouldn't work properly on the paperboards but actually worked in the real life, this is how their charm did it.
Well easiest trick is to tension the first curtain roller spring a little bit but this is a temporary fix which can cause uneven exposure afterwards. Postwar Contaxes obey a binary rule when it comes to CLA : "everything or nothing", and any exposure problem automatically leads to "everything" as for their maintainance.
I have three postwar Contaxes : an early IIa that works like a charm, a IIIa that now works well after a total overhaul, and a IIa which never ever wanted to work in spite of five successive overhauls (I am now keeping it apart in a box as a spare parts stock for the other two).
You will probably be advised to send your camera to a certain person in California who claims to be the only guy on this planet to have the skills to work on these cameras, this is just nuts and so, unless you want to pay big dollar and have your camera away for three years on a holy s**t "waiting list", try to find another workshop. Essex is good, you also have Eddy Smolov in NYC I guess.
The photo of the dogs was nice- too bad ; the last portrait of the girl can still be enlarged, the shutter blanking rathers look like modern fancy vignetting here... 🙂