Another possibility is the starting capacitor for the compressor motor is bad.

If the capacitor is bad then the compressor motor won't actually start, but the fan will still turn on. This will make it seem like the AC unit is running, but there won't actually be any cooling going on. The fans will still blow air, and it'll feel slightly cooler because it's moving air, but it won't actually be cooler.

Replacing capacitors is pretty easy and not too expensive. However, it can be extremely dangerous so I recommend you leave it to the professionals unless you have some experience working with electricity.