I live in an area that is subject to additional use tax. When I select the type of use tax, the amount doesn't change, although it should.

It depends where and how you would report additional use tax based on where you reside. It is a best practice to use your state's Department of Revenue Individual Income Tax Instructions to determine the amount of use tax applicable. 

If you had paid additional tax or had sales tax to report, this would be captured separately.

It turns out that TurboTax was using the wrong number to calculate the additional tax.  The amount did not change because the additional taxes calculated was too small to register (< $1).  TT used the previously calculated tax rather than the amount the previously calculated tax was based on.  The problem has been reported.