No harm in leaving it in gear. Leaving it in gear helps keep the rig in place when the rear stabilizers are off the ground and is one less step each time if you are doing a job that requires frequent repositioning of the machine (like digging out large roots.)
Pushing the tractor forward with the BH is a pretty common practice when trenching on level ground, but does require some skill and coordination -- it is important to push the tractor forward, not lift it up. In this case, you would leave the tractor in neutral. Ensure the BH swing is pointed straight back, not off to one side or the other. Same with the front wheels. Plant the BH bucket teeth in the soil near the tractor, raise the FEL bucket slightly of the ground, raise the stabilizers slightly off the ground. Raise the BH bucket as necessary to ensure all 4 wheels are on the ground, but not enough to lift the BH bucket out of contact with the ground.
Now gently and simultaneously raise the dipper, lower the boom and uncurl the bucket to roll the tractor along the ground. Basically the exact opposite of digging a flat bottom, and if you are not coordinated enough to dig a flat bottom then maybe get more practice in before trying the above. If you push down too hard with the bucket, you will raise the rear wheels of the tractor so it is balancing on the front axle pivot and the bucket -- no stability at all. If you lift the bucket off the ground, the tractor could roll away or back into the excavation. The trick is to keep the bucket teeth and all 4 wheels in contact with the ground at all times.
I would not try this on any sort of slope, nor would I ever attempt to reposition sideways. The first risks loss of control, the second risks damage to the swing mechanism.
But that's just my 2¢.