Turbo Tax is telling me that a state form will not be ready until 28 April. Won't this make my taxes late (past the 15 April deadline)?

If you MUST have that form, then it appears that you will need to file for an extension.  Be sure you send enough $$ with your extension request, so you definitely get a refund of excess.  Otherwise get stuck with penalty if a bit short.

Which state are you filing in and which form?

Many State returns have different due dates.  It depends on which state you live in. I know that Virginia and Delaware returns are not due until May. So check with your state.