They do. Usually, the question is framed in terms of the Bible Alone doctrine, which the Catholics reject. They way you frame the question as that of inerrancy of the Old Testament combined with skepticism toward the new is, of course, inherently un-Christian, so I would not expect it to be also a Protestant position.
As you well know, I don't do this.
As someone who has been through the shock of converting to Catholicism and having to deal with the evolutionism, higher criticism, and Biblical liberalism of the Church, I am sincerely interested at how other Fundamentalist converts deal with this problem. What I don't understand is pretending the issue doesn't exist.