I am finishing my second year, and I have used a laptop for every single one of my exams since I started lawschool. Its a huge advantage. And I guarantee you the teacher, whether consciously or not, favors the typed exams. Your hands never get tired, and you can edit your answers to your hearts content.
The school makes you install a program that locks your hard drive during the exam so that you cant access notes. If your computer crashes, your in trouble. It has never happened at our school. The software company says that its software cant cause crashes.
As far as using an actual typwriter. I would do it. You said you were concerned about not having the exam directly in front of you. Thoroughly read the exam and outline your answer ahead of time. Then all you need to do is glance at it here and there to make sure your on track.