IL does not require the same filing status to be used for federal and state return if one is a full year resident and the other is a part-year resident.  So, it's possible for file MFJ for federal and MFS for IL.

There is not sufficient information to make a determination because a lot of that depends on their level of income and the type of expenses they have for deductions and credits, which may or may not be allowed for MFS.

You will need to determine whether the spouse's move to TX is for only a finite period, which does not, generally, cause the domicile to shift and the spouse would then remain a tax resident of IL.  Once you have that determined, you can use PS to run some scenario calculations if the optimal position is not immediately clear to you.