Heat pump, Trane, late 80's vintage. To me a very typical heat pump. Air handler fan runs constantly. Turn up the thermostat, fan runs. Remove thermostat, fan runs. Disconnect the breaker fan turns off (duh). Reconnect the breaker with the thermostat high (no demand) the fan runs but the outdoor unit is off. Turn down the thermostat next to create demand, outdoor unit runs, air cools. Demand ends, outdoor unit shuts off, fan runs.
It is clear to me that the air handler fan relay is "stuck" on (or do you have another/better idea?). Where is the relay located? Inside the air handler or on the printed circuit board in the outdoor unit? Is there any good reason I cannot simply replace this relay?
Again to note... both units start, run, and cool when demand is present.
Thanks,

Unless the thermostat wires or the signal wires to the outside unit got damaged it will be the fan relay. The fan relay is located on the board inside IF there is a board on your No model number unit. If no board present it will just be bolted to the inside units wiring/service area. There should be a wiring diagram on the indoor blower access panel.
Some have said if the relay is located on the board if so equipped it can be replaced BUT good luck with finding the exact relay that is on the board if so equipped.

Thanks for the verification. There is a small control box inside the air handler. It contains the time delay and fan relay combo PN RLY02257A which it turns out has already been replaced once in 19 years. The only other components in the control box are the buzzing transformer, a fuse, and a terminal strip for terminating all the wires. I kept the failed relay and box. It has $70.25 written on it... YEECH!! I suspect that my web search will turn up the part for ~half that delivered.
Meanwhile I used the "hammer method" to fix the stuck relay (for a day or two? Wish me luck) which is working so far... one day and counting.
I have another 24VAC 40VA transformer here and may try to access a modern time delay relay and simply replace the obviously poor TDR and noisy transformer rather than spend more money on this electromechanical cluster eff RLY02257A.
Again, wish me luck.

See page 14 for wiring diagram. I am taking a guess since you did not post exact model number

http://www.aireng.com/clientuploads/Trane_Data_Guides/TWECD.pdf

GRAB this while you can it is cheap and free shipping

NEW RLY02807 (RLY02257) Time Delay Relay HVAC Trane - eBay (item 330355291149 end time Sep-25-09 10:04:12 PDT) (http://cgi.ebay.com/NEW-RLY02807-RLY02257-Time-Delay-Relay-HVAC-Trane_W0QQitemZ330355291149QQcmdZViewItemQQptZLH_D efaultDomain_0?hash=item4ceab5b40d&_trksid=p3286.c0.m14)

Also look down the above page for wiring instructions. Good luck.

THanks for the link to the product manual. Thanks for the eBay but it is now closed. I bought one this morning from Keith Specialty Store for <$22 with $5 for USPS shipping so I did pretty well. Of course I had to buy the newer version RLY02807 which is listed all over the internet as the newer/replacement for the RLY02257. So far the "hammer method" is still working and hopefully will work until Monday when the relay should arrive.

The new on will have a wiring diagram with it but if it does not there was a wiring diagram on the eBay sell site. Pay careful attention to the wiring since it is always a possibility it could change from the old to the new if you know what I mean. The deal you got was OK sorry you missed the eBay one. I think you will do OK replacing it yourself. Post back with your results.