Contacts inside the relay are problably kaput.And this is what they will do run on & off,just had the same experience with a fan relay not long ago.Buy a new one & keep it in the glove boxe for peace of mind.

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After looking at the old DME relay. it had a date code of November 1986.
can it be an original part for 1987. wow.
the new one i got was 993-615-227-00
old one says made in W. Germany(Like the car) and has the code. the new one i got was 944-615-227-00