Do you have a way to verify the voltage? This could help to more accurately assess the battery's condition.
Something else that comes to mind is the motor, specifically the brush's and commutator. Is the motor getting very hot to the touch? My theory is that the motor is worn or wearing out or the armature windings could be damaged too causing excess resistance. This can make an ESC thermal out or shutdown from an overload. If the brushes and comm don't look like they have a deep groove cut into them then cleaning with an electrical cleaner could help here. It's hard to say without being able to see the truck in person.
Have you verified that the drive train rolls freely? Did you remove the pinion gear and push the truck by hand? If not, try that to feel for tight spots in the drive train. Hold the truck in the air and rotate each tire and try to feel the wheel bearings at each corner. Frozen bearings can cause problems too.