Our sink that is part of a solid surface counter (not Corian, but similar) cracked.
I didn't see or hear the crack happen, just noticed it. Any suggestions as to how we
Can fix it without replacing the entire countertop?

To replace a solid surface sink will require a heavy duty router with a specific bit that , at once quides along the opening and reaches in below that point cutting in .
In short it gets cut out with a router , so you can glue one back into the nice neat hole

Call whomever you purchased the countertop from, most sinks are warranteed for 10 years. This is not a do it yourself project. It takes a special router and bit, plus expertise to do this without damaging the countertop.