Finding Handedness-Neutral Wording (Left and Right)

In times past, bias against left-handed people was conscious and acceptable. Lefties like me can breathe a sigh of relief that the cruelest practices against us have fallen into disuse and begun to seem barbaric, at least in Western culture. People are becoming increasingly aware that a bias of form still inhibits us to an extent: We encounter right-handed scissors, computer mice set up for right-handed people, musical instruments we find clumsy to use, and a left-to-right written language that can lead to ink smudges for us lefties.

Changes have been slowly happening on this front; but another, more entrenched problem remains unsolved: that of language. As the remnants of the older, more conscious anti-left-handed mood, our language keeps its warts.
Imagine this scenario, if you will. A man suffers a heart attack and is taken to the emergency room of a nearby hospital. An concerned party may ask, "How is he doing?" Some possible responses show the gravity of the situation I describe:
Response A: "He's all right now. The doctor said he should stay off his feet for a while and follow a strict diet. They're going to monitor his health closely."
Response B: "I'm sorry I have to break this to you: The essence of his life has left him. He wanted me to give you this if you didn't make it in time."
In Response A, 'right' is associated with something positive, a condition of well-being. In Response B, in contrast, 'left' is associated with something negative, the absence of something essential. More bias can be found examining the etymologies of words of non-English origin.
* SINISTER: 'Sinister' is the Latin word for 'left.' Unfortunately, it has come to be associated with malevolence in modern English.

* AMBIDEXTROUS: 'Ambidextrous' derives from Latin and Greek words and means having both hands function as right hands would. Not only does this etymology devalue the innate qualities of the left hand, it is at odds with facts: Ambidexters frequently have a slight preference for their left hand, not the right one.
* GAUCHE: 'Gauche' is the French word for 'left,' but in English it has taken the meaning of awkward and clumsy.
* ADROIT: 'Adroit' comes from French, too, meaning 'on the right.' In English it means clever and skillful.
'Right' also some nondirectional meanings: an adjective meaning correct and a noun meaning a legal freedom. The latter usage occurs too in French, with 'droits' being rights. E.g., "La declaration des droits de l'Homme et du citoyen" (The declaration of the rights of man and citizen).
I propose we abandon these insensitive usages and leave them for our history books. They all have synonyms that do not implicitly devalue left-handedness, except ambidextry, which will require a neologism like both-handedness.