Yes you need both fans.
When your AC compressor compresses the freon it builds heat. This heat needs to be removed so the freon gas will liquify, so they invented the condensor which is basically an AC refrigerant radiator. The condensor relys on a fan to pull air through it to be effective. If there is no fan there is no air movement through the condensor so it is not cooled. As freon heats, it builds pressure. If pressure gets too high because of overheating, the high pressure switch will disengage the compressor clutch stopping the circulation of freon, causing no cold air.
Get the fans fixed, it sounds like he switched your AC and fan relays. Buy new relays and replace them yourself.