Yes, the 5/8" drywall would help but I would run it the long way rather than paralell to the trusses. You may want to cut the first sheet in half lengthwise to offset the joints by a couple of feet.
I would rent a drywall lifting machine to press it up to the trusses and use a pair of screws (spaced 1" apart) every 8" across the sheet. The old will want to retain the sag and will have to be mechanically pressed up to get a satisfactory job.
I'm surprised the 3/8" hasn't given away under the load of insulation.