Hello, trying to fix central air system. I started with a short in my thermostat wire out by condenser. Originally I stripped and rewired, after triggering the contact multiple times unit shut off. Since then I've replaced capacitor contactor fuses for outside shut off and breakers and thermostat wire. Need help figuring out what next? When I push contactor in by hand unit comes on.

You are not getting 24 VAC to the contactor. Could be a bad wiring connection, bad thermostat, bad 24VAC transformer. Start searching backward until you find your 24 VAC source and then go forward.

If you have a voltmeter check the coil on the contactor, make sure you have 24V.