Make sure your wiring and connections are OK before replacing the probe.

I had a loose wire behind the panel.

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Now, the EGT fairly consistently acts like this: it reads "normal" on
cold start, taxi, runup and takeoff. Climbout it reads around 1200
deg F. like it always did. Then, after a few minutes, it starts to
fall, and within maybe a minute or two, the needle quivers at the
bottom of the scale. It never recovers from that; but at next flight
a week or two later the same behavior happens.