Using your voltmeter set to a scale that will read VAC, place your probes on the 2 largest wires at the bottom of the relay and confirm there is no voltage present. Capacitors store charge and although a whack from a capacitor will not kill you, it might startle you enough, you fall or otherwise hurt yourself. If the switch is always "closed", the start capacitor is always enabled, thus likely destroying the capacitor.
I am lucky in that I live near an electric motor rebuild shop that carries HVAC motors and run capacitors they will sell to anyone. Yes, it can get hot inside without HVAC but better to suffer through a hot house and get whacked by high voltage. Examples of motor capacitors are: You might have no choice but to pay someone to repair your unit. A dual "run capacitor" supports two electric motors, such as in large air conditioner or heat pump units, with both a fan motor and a compressor motor. This label will define what you must buy as a run capacitor replacement.