If the AC doesn’t turn off, it may be time to clean the condensing unit. Dirty condenser coils won’t give off heat efficiently and will keep the unit running.  Another possibility: The contacts on the outdoor run relay may have welded together—something that can happen over time because of frequent electrical arcing at the relay. Before checking the run relay, turn off the power to the furnace and the condensing unit. Next, disassemble the relay and pry apart the contacts. This should do the trick until you buy a replacement—which you should do soon.
We installed a TRANE XL18 HVAC with a TRANE HV80 furnace 16 years ago and it still works great in Tucson, AZ.  We inquired into a replacement for the future since the R-22 refrigerant will be off market at the end of this year.  Our maintenance folks want to sell me a new system anyway (even though the TRANE works great).  We are considering the BOSCH Inverter heat pump as a lower cost option down the road.  Watch out for the tricky salesmen trying to sell beer at champagne prices though. Get estimates on whether you need to replace your current system and how much it could cost.  And if they are factory certified.
Some systems include an "economizer mode", which is sometimes called a "free-cooling mode". When economizing, the control system will open (fully or partially) the outside air damper and close (fully or partially) the return air damper. This will cause fresh, outside air to be supplied to the system. When the outside air is cooler than the demanded cool air, this will allow the demand to be met without using the mechanical supply of cooling (typically chilled water or a direct expansion "DX" unit), thus saving energy. The control system can compare the temperature of the outside air vs. return air, or it can compare the enthalpy of the air, as is frequently done in climates where humidity is more of an issue. In both cases, the outside air must be less energetic than the return air for the system to enter the economizer mode.
If your compressor doesn’t have an overload button and you hear it humming or buzzing, poke a screwdriver or stick down through the top grille and try to spin the fan blades clockwise. If doing this gives the fan enough of a boost to get it going, the unit has a faulty capacitor that must be replaced. See How to Test and Replace an AC Run Capacitor, above.

"Customer service was great, and the price was in-line with other places, and fast service. We had a whole replacement of the outside and inside. I would have liked to see a better install for the furnace and new thermostat - the new thermostat should have gone where the old one was, and even tho the new furnace was a tight fit, they didn't offer to come back and repair the case molding, or give me a discount on the damage done, and I believe they should have put a disconnect box on the outside of the home, to make it easier for any future repairs needed. Other than that, am very happy with the cooling and heating!"