I want to use a different address to apply for my state refund

TurboTax will use the mailing address that you entered in the My Info section for both your Federal and State returns. 

If you are filing your Federal return first and your State return later, then you can update the address in My Info. 

You can file your federal return and skip the state, but your e-filed federal return will need to be accepted before you can e-file your state (or, you can just paper-file your state return later).

Here are two ways to skip your state in TurboTax Online:

1. Delete your state return, file your federal return now, and then come back later to re-do your state return.
2. Pay for both your federal and state returns now and when you get to the Here are your filing options screen, select Change next to your state return. On the Change filing method screen, select I'm not ready to file and Save.

Method 1 doesn't require up-front payment for your state, but you'll have to re-enter your state information later. Method 2 saves the work you did on your state return, but requires up-front payment.

See this TurboTax tips article if your address has changed and you need to change your mailing address with the IRS.