That little red button you keep pressing is know as the high-pressure cut out switch. It is designed to stop the unit from working under high pressure conditions that will ruin your cooling system if left to run under these conditions.
Chances are your condensor fan motor is bad (or going bad if still turning somewhat) and not pulling enough heat from the system, causing a high pressure condition. It could also be the fan motor capacitor not starting the fan and that would be a very cheap repair. A new capacitor will come with a new motor, just be sure if you buy the capacitor separate, that is has the same specs. I don't know if your system has a variable speed fan controller but, I've seen these go bad as well. This controller adjusts the speed of the fan so you don't cool it too much while running the A/C in cold weather.
You should be able to replace the capacitor for under $5 but, a new condensor fan motor might cost you around $250.