It sounds like you are low on freon.
Another symptom of this is that the unit runs continuously except when in a defrost cycle re: it is trying to reach the temperature set point but cannot due to low freon (poor cooling).
Here is a link, take a look at the example pictures part way down the page.
The only proviso on checking the coils is that you have to be sure that the unit has run a couple hours without a defrost cycle.
If it is low freon it will be an expensive repair as you need a pro. Also there is no guarantee that they will be able to locate and repair the leak.
Often it makes more sense to put those dollars towards a new unit.
PS: You said the fan was working but did not specify which fan so also check that the evaporator (freezer) fan is running.