Adding floor joists to an attic
It sounds like the existing 2x4’s are 24” oc and span 13’. (The truss has nothing to do with the existing ceiling or future floor of attic.)

Yes, the existing 2x4’s can barely carry the existing ceiling material. 

To create an attic floor, I’d add 2x6’s at 24” oc spanning 13’ to support about 25 - 30 lbs. per square foot, depending on the species and grade of lumber. (This is good for Christmas decorations, luggage, etc.)

For more support, I’d add 2x8’s 24” oc spanning 13’ to support about 40 - 45 lbs. per square foot. (This will support books, people, etc.)

I doubt the existing walls will need to be modified, if they are bearing walls now.