If the contactor coil is AC, switch wires won't help. I don't believe the solenoid is pushing out, it's just not pulling in. This could be a bad coil on the contactor solenoid or the compressor is being held out for other conditions.
Other conditions could be; bad T-stat calling for cool and heat at the same time however, one is suppose to take precedence; low ambient temperature or it could be locked out on low pressure. Check the freon pressures (hi/lo) on the unit and maybe use a leak detector to check for leaks. If this has service ports on the fridge lines, those are highly suspect as well as where the fridge lines enter and leave the condenser coils, same may apply to the evaporator lines/coils but, the compressor coil lines are more typical. You could temporarily bypass the pressure sensor to test it but, don't run it for long (a few seconds tops). However, this will only isolate the problem to pressure if the compressor comes on w/the sensor bypassed. You'll have to determine the system pressure to know if the pressure sensor is bad or whether the system is low on charge.