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Any decent HVAC contractor will do a load calculation for free, don't guess on this one.
Spend all that time and money and then not be comfortable in your own home. Find someone
Who will put in...

The 13 seer law is a federal law. I would definitely replace entire
System. Always get at least 3 bids when purchasing a unit. And if a company is afraid to use the R 410a refrigerant, find another...

After checking breakers, I would check Indoor unit float safety switch, sometimes installers
Use the float switch to shut all power to thermostat off and others shut down outside unit only.
If...

It does sound like your contactor is stuck shut. If you want to know
For sure it's the contactor and not 2 low voltage wires grounded toge
There, you can turn off both breakers to the unit, normaly...