I mailed in my state return without my federal return. What should I do? Remail them together or mail the federal return separately, or something else?

Were you suppose to attach a copy of the Federal with your state?  Wait for the state to ask you for it.  Did you also file federal to the IRS in a separate envelope?  Federal and state go to different places.

you mail your 1040 to the IRS.

you mail your state return to the address they give you for mailing. (done already)

These are two separate action.

See the IRS website to get the correct address for the state you live in.

Note IRS is way backed up on mailed returns so it will take a long time to hear anything.

At this time, do absolutely nothing. Otherwise, you risk making things worse. If the state needs a copy of your federal tax return, they will send you a letter requesting it. If you "jump the gun" and mail it to them now, chances are when they receive it, they'll have no idea why you sent it and will just trash it..... thus delaying things even longer. So just wait.