Radiator Fan relay sticking - '86 4kq

[Dave Conner]

Dear Wise Audi Guru's,

Why does my radiator fan relay stick in the on position?
I replaced the relay several times, but always with a used one.  The most
recent occasion was several months ago when I replaced the relay and the
problem seemed cured... until yesterday.  I got a call for help from my
daughter because the car wouldn't start.  I went and found a dead battery,
swapped a good one and immediately heard the roar of the fan.  A light tap
on the relay shut it off.
Of course the fan doesn't ever come on much in the cold weather we've been
having these last few months. The fan itself seems OK with no unusual
noises, etc.

I read suggestions in the archives that the old/original fan may have
increased electrical resistance to the point where it draws higher current
that welds the relay contacts together.  
Is there a way to test the electrical resistance of the fan?
Should I just assume this is the case and replace the fan? 

Any tips or BTDT appreciated.
Dave C.

Columbus, OH 

89 200Q 
89 100
87 5KS
86 4KQ