Skipping is a real good reason to think the chainring is worn to the replacement point and your picture of the crank seems to show many of the teeth are worn to a sharp point or noticably "hooked" so the rings are indeed ready for the dumpster. A cassette with truly broken teeth on one or more cogs is also DOA. When you replace them, be sure to buy a new chain also.

Out of curiosity, how many miles are on these items?