Can't get my AC to turn on at all
Sounds like a blown compressor.
Bad contacters don't usually pop fuses. And if it was simply the fan not coming on, the compressor should trip its own over-heat, not your breakers (unless it's old and grumpy: same prognosis; dead compressor).
If I were to keep trying to fix at that unit, the first thing I would do is install a &quot;fused&quot; (circuit breaker) disconnect for the condenser outside and a convenience outlet (both are now required by code).
That's step one of installing another unit, so it's no big loss if that doesn't help the diagnostics.

Air Conditioner Not Cooling : Tripping Breaker And Blowing Fuses –YouTube

My suspicion is we've got a bad compressor. Let me go grab my multimeter and get this thing tested out. Again, I'm going to go ahead and discharge the capacitor and then pull all the leads off the compressor. Then I'm going to go ahead and leave the fan motor connected and test. If this doesn't blow the breaker then that means our compressor's bad and it's time for either a new compressor or as old as this unit is, it may just be time for a whole new unit.
... It is as I suspected; the compressor's got a short in it. We're going to need either a new compressor or a whole new condenser unit. If I wanted to, I can recycle the fan top, the fan motor, and fan blades. That's a little bit more advanced troubleshooting and proving that it's actually our compressor that is down. We can replace the compressor if it's a clean change out. On this one, I think it's probably just maybe time for a new unit.

I've replaced compressors on commercial units (whose construction is conducive to such); it's not a very common practice for split systems, because that's  the same amount of work, and you'd still have old equipment throughout.