[COLOR="Blue"]Am I correct in guessing this is the problem?[/COLOR]
Yes it does sound like a sealed system problem.
I am assuming that the evaporator fan runs and the compressor is on all the time other than when it goes through a defrost cycle.

Since this will be expensive, be sure that the compressor has run for at least an hour before checking the evaporator coils. If you check them shortly after or during a defrost cycle you could come up with the wrong diagnosis.

If anyone has had this repaired, about what did it cost? Trying to weigh the cost of repair vs replacement.
I do not know the cost to repair in your area but probably a minimum of $500.00 and could be considerably more.
There is no use just adding freon, they have to locate the leak and repair that. They may not be able to find it so you pay for the service call and end up with it not being repaired.
Also you will probably need a new drier at minimum and it may have screwed up the compressor since there is now probably air in the system