I'd replace the sink with a like sink. Assuming you can find a source. Corian dealers don't typically like to sell to DIY folks. Need to be certified and all that. But, if a sink can be found along with the correct adhesive it's doable.
Cutting it out can be done with a slot cutter in a router from the top. Use one with a guide bearing. May be difficult to find one with the necessary depth of cut but they can be purchased. Stick the new on in place with the Corian adhesive, that's how it was done when first fabricated. Clamp using a bridge from the countertop and through the drain hole to hold it in place until cured.
I was a certified installer for a similar product, not Corian but similar, before the maker got out of that market. Real stone became cheaper than the imitations!
Look around here:
https://www.solidsurface.com/adhesive.
On second thought, for a quick fix, grind the cracks out from both sides, don't grind so far that you crack a gap all way through. The order some of the adhesive from the above site, with applicator. Fill the cracks Sand to an acceptable appearance. About a hundred bucks for the tool and material.