'87XJ aux fan stays on after warm-up when cold outside

Long explanation with all the details ... perhaps a simple situation ??:

Daily-driver '87XJ Limited 4.0L in very good mechanical condition with auto-trans and A/C with ~ 313K miles (but A/C has not blown cold air for ~ last 2 years and I have not fixed it since it does not bother me) ...

... used in NY/NJ area... currently in the cold-weather situation over the past few weeks...

... changed the lead-acid battery ~ mid-December last month (I have changed the GRP 58 battery every 4 to 5 years since I owned the vehicle)...

... since Dec 2009 battery change the odd behavior is after warming-up on the highway (regardless cold weather AND coolant temp is NOT overheating) the aux fan comes on and does not shut-off...

... first I thought it was the aux-fan relay (which was ~ 3 years old) so I changed the relay last week but aux fan still stays on (after warming up on the highway) ...

... today the aux fan stayed on even after I turned-off the ignition key (what is that about ???) so with the fan running and the ignition switch off I pulled the relay (the fan stopped, of course) and then immediately re-inserted the relay and the fan did not run !!! ??? ...

... I have owned and driven the vehicle as a daily-driver since Sept '97 over the past ~ 200K miles and it is very reliable (with my weekend fix-it projects) ... I changed the radiator twice since I've owned it (first to a 3-row and ~ year and half ago to a 2-row)... when I had first changed it to a 3-row I had also bought and installed a new temp-sensor for the radiator (input to the aux fan relay) and then re-used the temp-sensor into the 2-row ....the Bosch water-pump has ~ 72K milles on it (probably due for a change before next Summer) and there is no coolant leaks anywhere... also, since it's Renix, I have made sure that there is currently no "air pocket" in the cooling system... but yes, the serpentine belt has ~ 80K miles on it (I should have replaced that last Fall 2009)...

... from what I read from this link (found via another post)
http://www.lunghd.com/Tech_Articles/Cooling/Electric_Fan_Diagnosis.htm
it is either the A/C (which I am NOT using) or the radiator's temp-sensor that provides the closed-circuit into the aux fan relay... but the diagram looks like the ignition switch needs to be in the on-position to complete the circuit into the relay and over the past summers when the aux fan was on (due to summer heat) the aux fan would then immediately shut off as soon as I turned-off the ignition switch ...

... but today when the aux fan continued to run after I shut-off the ignition switch... that thru me a curve ball (I sort of laughed a little bit)...

... any ideas and especially any prior experience with a similar situation would be appreciated...

thank you.

The only thing that could keep it on after the engine is turned off is the relay. I don't believe the temp switch in the left rad tank has power with the key off but maybe you've got power staying on to it with the key off.

It's possible the fan is drawing too much current messing up the contacts in the relay.

thanks for the feedback..

... yes... it was my first thought that it was a bad relay... which is why I changed-out the 3 year-old relay ~ 2 weeks ago to a brand-new relay... but the odd behavior is still happening with the new relay (as I had wrote)...

... note: I use the "water-proof" relays from PartsExpress (as opposed to a lot of opinion that the Bosch relays are the only way to go)... I use the PartsExpress water-proof relays almost everywhere under the hood since after I installed air-horns (few years ago) up underneath the base of the radiator (there is a nice empty cavity there for an air-compressor and 2 horns) I have found that the PartsExpress water-proof relays last longer than the others especially when the relay is installed low-to-the-ground

Here's the follow-up (it may be interesting to some Renix XJ owners):

couple of weekends ago I un-bolted the aux-fan relay-base from the inside of the ds fender and turned the base upside down.

I did this to cut and tap-into the red-trace colored wire ("hot out" from the relay #87 connect) to add-in a parallel wire to a new small red indicator-light that I added to the interior dashboard area
- so the I would get a dashboard visual when the aux-fan was having power-in.

Interesting point #1 of 3:
after twisting the relay-base upside-down it was clear that the insulation on both the relay #87 red-trace hot-out wire and the insulation on the relay #85 (Trigger-in) orange-trace wire were both slightly worn-thru and a small-patch of bare-wires for #87 and #85 were touching each other.
OK... so this is why when the aux-fan relay turned-on from initial current thru relay Trigger-in #85 that the aux-fan would never turn-off until I pulled-out the relay (that is: once the red-trace #87 power-out wire had power, it would then be a constant-feed into the orange-trace Trigger #85 wire... an infinite-duration Trigger-in).

After wiring the dashboard new red indicator light into the re-insulated red-trace relay #87 wire, then interesting point #2 occured ... only a highway speeds.

Interesting point #2 of 3:
with the new indicator light wired-up and after I got up to highway speed (especially after getting up past ~ 60 MPH), surprise was that the indicator light was glowing slightly... not a "solid-on" bright-red light but a glow that would decrease if I slowed-down (and then the light would not glow at all under ~ 50 MPH)... but would glow again if I sped-up ... hmmm !!
I then realized that the dashboard indicator-light was glowing relative to highway speed due to the aux-fan blades must be turning due to air-flow in-thru the radiator and the aux-fan motor was behaving as a current-generator producing a small current into the red-trace wire's indicator light.

Interesting point #3 of 3:
and then it occured to me as to why the original problem that I reported happened ... why the aux-fan was initially turning only at highway speeds in winter-time temps (eg: below 25F) with no cooling-system overheat.
The aux-fan was turning itself on !!
- via producing enough current out into the red-trace wire which then flowed thru the worn-thru insulation into the orange-trace wire Trigger-in relay #85 which then closed the relay power-on internal contacts which then directly-powered the aux-fan for an infinite-duration (until I would pull the relay out).

As stated above:
perhaps some Renix XJ owners will find this to be aux-fan relay trouble-shooting info that they can use some day.

Another great example of how clever electrical Gremlins can be!!!!