You are correct in both your assertions that we never find out exactly what is causing Betty's shaky hands, and that we know based on evidence that it is likely psychological.
As you mention in your question, the shaky hands are a part of Betty's character from day one. The first observance of this problem by the viewer is roughly three months after her mother dies (by Betty's own admission at her first visit to the psychiatrist). The issue is forced in "Ladies Room" (S1E2) after two hand-shaking incidents (the first was following a night of heavy drinking, the second when Betty's hands freeze up while driving sending the car up a curb) Betty is encouraged by her doctor to see a psychiatrist, as he believes the problem could be psychological.
At the conclusion of "Ladies Room," we observe Don listening to the psychiatrist's report on Betty's visit, and then nothing. After this episode, the writers leave the viewer to draw his or her own conclusions. Betty's shaking hands largely stop. But in the S2E3 ("The Benefactor"), Betty is hit on by a married man while at a horse stable, and in the process of lighting her cigarette while rebuffing his advances, we see her shaky hands return.
No, we do not find out precisely what causes Betty's shaky hands. Yes, it likely is psychological based on circumstantial evidence. No, there is no overwhelming evidence that there is something physically wrong with Betty.
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