Faculty award GRAs in order to get project-related work accomplished. If you are interested in an area at the department you want to attend that has external grants and contracts, attempt to get a GRA position. To do this you need to identify a faculty member in the department that does research in the field you want to study. Typically the Web is a big help in doing this. Then contact the faculty member.
Email is a good way to start.
If you can find a faculty member at your undergraduate university that knows the individual with whom you would like to work, see if he or she will call and speak of your interest and capabilities. This can really help to give you a leg up on others that are after the same position. While it is time to be concerned, it is not time to panic. All departments make GTA awards just before the Fall semester begins. New grading and laboratory assignments happen for many reasons.