Weird noise / feel when pushing in the clutch while moving
venom
Jeeper
I'm surprised, nobody has mentioned clutch or throw out bearing. These wonders of the mechanical world of jeeps are generally 100% correct though. Maybe an odd feeling in the clutch along with noise when the CJ is running is all about the motor and Transmission mounts. No matter what replacing yours would not be wasted money, your engine most certainly is to far back.
Good golly, more to think about

I would suggest replacing the two front motor mounts first. You will have to move the motor forward to position the mounts up. I was able to push it forward with 2 bottle jacks at an angle pushing the engine up and forward. An engine hoist will work also. If you get the ones from m.o.r.e. it is very noticeable of just how solid those suckers are. I have tried replacing with both the rubber and poly mounts and think the solid mounts are the way to go.
The solid mounts are on the way. Got the shipping notification just a few minutes ago. The last time I replaced the mounts, which is subject to my memory again, I just used a few jacks. I don't have a cherry picker or any other hoist at the moment, but if I need one such is life. I'll go with what I have at first and go from there.
with HH and T'head.
Loosen the tranny mount to help with this movement.
Don't forget the bushings in the clutch bellcrank. BTDT
LG
Bellcrank bushings! Argh! I'll have to order them, because I didn't even know they should be replaced. At least they're relatively cheap. I've only found plastic ones, so I assume that's all there is. You guys are a wealth of information!
Thanks again!