Should I register at the hospital before I am full-term?

Yes. Hospitals will be billing for the service they provide to you. It is very helpful if you preregister so that you are not trying to fill out paperwork in between contractions. This also helps you avoid mistakes with medical billing that you will have to fix after discharge when you are busy with a newborn at home.
Yes. Most of the ob's I work with have their patients pre registered by mid pregnancy. Having your information in their computer system & crosschecked & verified before you need to go in helps you. Some will need to go in to the labor suite on more than 1 occasion to monitor fetal condition & check for early labor. Simplifying the registration will smooth out the process for you. Ask your ob.