Did the evaporator coils have a fairly even coating of frost?

If I interpret the post correctly the compressor is cycling more or less as normal.
Try setting the control thermostat to a colder setting.

[COLOR="Red"]Did the evaporator coils have a fairly even coating of frost?[/COLOR]

If the compressor is cycling more or less as it always has then it is a temperture control problem.

if the compressor runs all the time except when the unit goes through a defrost cycle then it is most likely a sealed system problem depending on how the evaporator coils looked.

[COLOR="Blue"]The coil had no frost on it. It was cold but not frosty.[/COLOR]
If the compressor had run for about an hour before you did the evaporator coil check and the coils had no frost on them then it is a sealed system problem.
This will be an expensive repair and often it makes more sense to put the dollars towards a new unit.

It is important that the check be done after the compressor has been on for at least an hour. If you do the check shortly after or during a defrost cycle then you could end up with the wrong diagnosis.


[COLOR="Blue"]Could the thermostat be broken in the closed position? That is, never turning the defrost heater off?[/COLOR]
Could be but the compressor is running so the unit is not in a defrost cycle.
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If I were to dis connect the defrost heater, would the unit work properly and get colder then?[/COLOR]
No
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Or if the timer was not working properly have anything to do with it?[/COLOR]
No.

Everything depends on the compressor if it and the fans are on all the time except when the unit goes through a defrost cycle when they are all off then there should be frost on the evaporator coils.

If I interpret your post correctly then your compressor and fans are running.
I am assuming they run all the time between defrost cycles.
So I think you have a sealed system problem.