I transplanted everything but the board into that machine. It has the same symptoms: time failing to count with MODE32 enabled, counting a second about every 3 seconds with MODE32 disabled.

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Did "everything" include the pram battery? Either way, if you have a voltmeter, measure the battery voltage in operation. If it reads ok (say, 85% of nominal or better), then you can rule out the battery as the source of trouble.