Yes. You can file them separately. Although state returns can be e-filed with your federal return (or after your federal return has already been accepted), it's no longer possible to e-file state returns before the federal.
Can I file my federal taxes now and skip my state for later?
You can, 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:
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.
You cannot e-file anything right now. E-filing will not be open for federal returns until late January. Even if you manage to e-file sooner, your return will just sit on a server where you will be powerless to fix any mistakes you discover. And...if you are getting earned income credit or the additional child tax credit, the IRS will not send your refund until almost the end of February, so best advice is to wait. Wait for all of your documents, wait for your forms to be ready, and e-file all of it at the same time. We see people who file federal, forget to file state, etc. etc. etc. Rushing rarely leads to happy results.
@Lizz913 - by law, the IRS can not process a tax return with EITC or additional child tax credit until Feb 15, so no need to rush....better to wait and get the tax return correct. these boards are chock full of people who rushed to file and then were forced to amend when they realized they had a mistake.
Currently amendments appear to be taking 6-9 months to process, so wait until you are absolutely sure you have received every document required to file!