I've paid to e-file my state and federal returns. My state return rejected. What can I do to e-file only my federal return?

Your federal return should not have rejected because your state return rejected. Are your sure your federal rejected rather than being accepted? You can confirm the status at this link: https://shop.turbotax.intuit.com/efile/efile_status_lookup.jsp

If your federal return rejected, it would have been for some other reason. The rejection email would have told you what the reason was and how to fix it. 

Incidentally, you don't pay anything extra to efile your returns when using the online product. The fees are the same whether you efile your returns or print and mail them. The fees are for the use of the software to prepare your returns.