It's hard to say without more information or pictures...if the cracks are only on the surface, and don't go clear through, then you should be able to fill them and paint for a temporary fix.

What I'm wondering more, is this: if the sink is a separate piece from the countertop, why not just put in a new one without replacing the entire cabinet?

There are companies that will "refinish" tubs and sinks. But you also buy paint designed for the same.

Not sure about the idea of painting flowers - what kind of paint were you planning to use?

Here is a link that might be useful: FOR DUMMIES REFINISHING EPOXY KIT

You can buy an entire vanity and top at Home Depot for $40. It won't be the best quality, but it's the cheapest solution that will actually work.

All good advice...thanks. I did find a few how-to's on the web. They involve a lot of epoxy, etc., but a few looked doable...I'll think on it.

Replace it.