Since you have limited time, and pay by the hour, your best approach is to get as much info as you can before going to the registry.
So using your GrGrandmother as an example, if she died in the 1960's, you wouldn't find her death information in the Archives. The Archives has records through 1915. Her record would be at the Registry. You would look her up in the index to identify the record location, and then go lookup the record. But to Tina's point, there might be more information available online in the Social Security Death Index. The SSDI is accessible though Ancestry. If your GrGrandmother had an uncommon name, it might not matter, but if it was common, SSDI could help you narrow down which record was hers. For example, if her name was Mary
Smith, you might have to look at lots of indices/records until you came across the one that's yours. The SSDI can help narrow it down since there is often birth date information listed as well. Once you know when she died (lets say April 1965), it can optimize your lookups at the registry.
To second Tina' comment, don't be afraid to ask for help.
John