I\'ve seen the following thread which is related to my question:
WPF ComboBox: background color when disabled
The above deals with changing the Content Temp
For this situation I like to set Focusable=false and set the background color to my desired value (in a data-bound trigger). This is maybe a little hacky, but so is rewriting the control template for the whole TextBox. An alternative to Focusable is IsReadyOnly, but that doesn't work for as many controls. It does ensure the caret disappears, though.