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I recently bought an '01 c5 w/ low mileage (17k) and I have noticed recently, that while driving, it is difficult to move the shifter into each gear (there is a short stutter/lock up while shifting into gears).
It seems that this is still present when the car is off but it's not as pronounced.
Remove the shifter boot and the rubber close out from the top of the tunnel.
When you SHIFT, examine the shift box and see if it moves during the shift.
It shouldn’t move (at least not much at all) The box is mounted in rubber bushings. Two bushings on the top and one long solid rubber lined V notch on the bottom.
See if the bushings are deteriorated or if the four bolts that hold the shifter plate to the box are loose.
Look at the coupler that connects the box shift shaft to the trans shift rod. It should be 100% tight and have ZERO lost motion.
Also check your clutch fluid and make sure the fluid is clear and at the proper level. If its nasty and low, you may not be fully disengaging the clutch. If thats the case, it will QUICKLY wear out the synchro.
If its NASTY, I recommend properly flushing and bleeding the entire system. The slave and master cyl with new brake fluid.
its normal these cars have notchy shifting with the stock shifter box. Some are more pronounced than others. I would change the trans fluid with some good synthetic fluid and see if it gets better, if not try the tips above and upgrade to a better shifter.