Should I install GFCI outlets or run new grounded cabling in my older home?
Replace the outlets with GFCIs -- the operation of a GFCI in no way depends on the presence of the equipment grounding wire. You'll have to use the "No Equipment Ground" sticker that comes with them, by the way.

P.S. on the metal boxes -- since it sounds like your house is a mix of K&amp;T and other wiring techniques (perhaps NM additions), you cannot ground to the boxes -- that technique only works if your house was wired with BX/AC, MC, or metallic conduit (RMC/IMC/EMT).

(Confirming that you have K&amp;T and NM mixed in your house, based on your photos, by the way -- I also do not see a ground wire on the NM, but the angle of the photo makes it hard to tell.)