I would be really surprised if the shaft packing would cause that symptom.
Like Troy said, the prop shaft uses the water as lubrication so it is a really BAD idea to run the prop on the trailer. The shaft packing may tolerate it for a while but the cutless bearing could get destroyed.
The cutless bearing is the "bushing" that is in the strut, which holds the shaft just ahead of the prop. In normal operation the strut is completely underwater. Groves in the bushing allow water to flow down the length of the bushing and creates a film of water over the shaft as it turns.
When you operate it dry the friction will quickly heat the bushing, make it expand, grip the shaft tighter, bind, make noise, tear itself up, etc.
Try turning the prop by hand. You should be able to turn the prob with only a minimal amount of resistance. It should also turn evenly throughout the complete revolution. If you have to really lean into it to get it to turn then something is very wrong. If the shaft turns freely for a bit and then binds then it indicates that the shaft is bent or out of alignment.
It doesn't take much of a bend on a prop shaft to create so much vibration that it can seriously damage things. Generally, however, the boat will still run and perform, it just will shake itself apart.
If the shaft is bent so bad that it is binding to the point that it is dragging down a 300 HP engine then I would expect it to be fairly visible to the eye. Turn the prop and see if it wobbles or if you can see any run out in the length of the shaft.
If you are not sure then pull the shaft out and take it down to a shop and have it tested.
It is also possible that something really bad is happening in the transmission. If the clutch was slipping it could fool you into thinking that the engine was bogging down.
Pull the dipstick on the transmission and smell it. If it smells burnt then just give a heavy sigh and repeat: BOAT = Break Out Another Thousand!