(I am assuming they still use a cable style clutch on the 04 v6)
It sounds like your clutch is not fully engaging. If that’s the case the burning smell would be your clutch disk slipping and overheating. As far as the hard shifting, that can be caused by the clutch not disengaging. Because you have both symptoms it seems like your clutch is stuck somewhere in between engaged and disengaged.
This could be caused by several things. A mechanical problem with the throw out bearing, clutch quadrant, cable, or pressure plate.
Some things you could check at home
-With the car on jack stands have a friend depress the clutch pedal while you look at the clutch fork where it goes into the bellhousing. The movement should be smooth and about 1.5”-2” (I think). You may have to remove a sheet metal cover over the clutch fork in order to see it
-Use a flash light to look under the dash. Inspect the clutch quadrant to see if there is a mechanical failure or abnormal ware. ( it is a half circle plastic piece near the steering column)
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If the cable is moving freely through the full range of motion then I would guess that the problem is in the pressure plate.
If the cable is not moving the clutch fork through the full range of motion and the quad rant is fine than the problem is most likely with the throw out bearing.
**** I do not recommend that you drive the vehicle at all if possible**** further damage to the clutch can occur from overheating.
Good luck