I recently moved into a new house which accepts a 4 prong dryer cord. My dryer, a Kenmore 110.66922501, is currently wired for 3. As wired for 3 prong, the neutral connection on the dryer has an additional green (with yellow stripe) wire attached. This green striped wire appears to terminate at a point on the chassis. When changing to the 4 prong cord, should the green striped wire be moved to the chassis to be connected with the cord’s green wire?
Yes, that is the most important thing you need to do!
Neutral is a power wire that handles return/balance current. It's a "hot" wire, but is near ground voltage under normal circumstances. However it can be "hot, which is why it is insulated.
On a proper and modern 4-wire dryer connection, we want neutral connected to to the electrically active parts of the machine, and the chassis attached to safety ground.
On a 3-wire dryer connection, they made the decision to bond the machine chassis to neutral. It was known to be unsafe, but was considered less unsafe than doing nothing at all, on the rationale that the wiring is rarely disturbed and the neutral is unlikely to fail (electrifying the chassis of the machine if it did).
So yes. As part of a conversion to 4-wire, you really want to remove that bonding jumper.
It should go somewhere secure where it won't flop around and hit a hot wire.
You can only put 1 bare wire on a screw, but if all wire ends have ring terminals crimped on, you can stack those.