I couldn't get the video to work properly so haven't been able to see the incident you are referring to.
But, perhaps we should take the following into account.
Camilla might, just might, like many others be embarrassed about needing help to get in or out of cars and have asked not to be assisted either by her husband or by an equerry.
Not a good idea, as it sounds rather odd for any husband just to walk off regardless of whether his wife is actually with him or not.
I doubt it has anything to do with proctocol, as royal proctocol still states firmly that the sovereign takes first place whether he is offering his arm to his queen or not, so there is no need for the king to have walked off alone.
The Duke of Edinburgh could not walk beside Queen Elizabeth II because she was queen and he as a prince and duke ranked below her,
A queen consort is like any other married woman inferior in the table of precedence to her husband. (Don't eat me ladies, I didn't make these rules.) So she can walk beside her husband instead of three paces behind him.
But hands in pockets are definitely not good manners!
If they are cold: wear gloves!
Whoever you are, all boys of our generation, and His Majesty is only 3 years older than I, were taught it was the height of bad manners to keep their hands in their pockets in public.
The reason never dawned upon me, but DH kindly explained to me, when we were newly married, that boys or men with a hand, or both, in their pockets were commonly said to be playing with themselves. Thus the embargo on doing so in public!
Female, wifey-woman as I am, I had not realised they could actually reach the penises from their trouser pockets.
I don't doubt his explanation, but it doesn't explain why we girls were not allowed to keep our hands in our pockets, as no-one ever told little girls of my generation that we could play with ourselves in that sense!
Giving people your left hand to shake is also rude, so if the Queen has a sore hand, she should refrain from shaking hands at all, but just smile and bow her head a little.
As it is the prerogative of a lady, and of the person of highest rank, to decide whether to shake hands or not, she would be doing nothing wrong by not shaking hands.
But that said, could we not just give King Charles and Queen Camila a little more time to get used to things?
Or perhaps get her dresser or lady-in-waiting to suggest substituting a shoulder-bag over the left shoulder for the clutch bag?