We live in Arizona, married, same household with 2 children. can we or should we have filed separately?

It would be rare for filing separate to reduce a tax burden as the tax rate is higher.  You can create two separate returns, under new account ids and passwords,  for each of you to see how it works out.  You cannot just change the filing status on a joint return to separate.  Remember, if you file separate, you have to both itemize your deductions or take the standard deduction.

In general, there are more tax benefits when filing married filing joint. However the FAQ below will give you much more information to make an informed decision that will be right for your particular situation.

https://ttlc.intuit.com/replies/3288477

You can but you will not be able to claim certain tax breaks given to you for your kids.