The other problem, if you look at the typical social skills advice, is that you're supposed to maintain eye contact during the handshake - making it even more uncomfortable. And, as someone who is often anxious, I get really self-conscious about my hands often being quite sweaty.
The worst for this is a certain kind of person, usually over-confident, arrogant business people, who believe that how you shake hands is some kind of personality test that they will then use to judge the other person. I've met quite a few managers and sales people who, after I've forced myself to go through the whole pointless ritual, then come out with something which sounds like - "That wasn't a proper handshake, do it again, this time with "feeling"'. Eh, what? I thought the objective was to say "Hi", then negotiate an objective business agreement, not engage in some kind of dick-measuring contest based on grip strength (when I say "dick-measuring" - I have noted that I've never encountered a woman who feels the need to analyse my handshake this way).
I have been known to get my own back sometimes by REALLY squeezing the next time around. I might be all skin-and-bone, but I've played bass every day for thirty years and used to be a rock-climber. If they complain that it hurts, I just tell them, "You just told me my handshake was too limp, I was only doing what you suggested."
In my experience, there is a correlation between this kind of stupid "handshake test" and people who have been through a certain kind of business/management training - the sort that likes to get employees to sit through Myers-Briggs personality tests, and all that other pointless rubbish that real psychologists and sociologists dumped years ago as too simplistic, unreliable, and unrepeatable to be much use for anything.
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When you are fighting an invisible monster, first throw a bucket of paint over it.