Is there a proper way/order to delete all the images, then the gallery without affecting the other? Is there an "original" gallery?
I have seldom done this, but here are a couple of tidbits. Before starting, make sure when you are logged in you can go to "Contents" for each of the two galleries and see all the images showing so you know both galleries are properly configured.
If you edit an individual image in a gallery (talking the newer "drop and drag" editor here ...), toward the bottom of the main edit frame is red text: "Delete image from: " then a pull-down and after that a button for "Delete Image." The pull-down shows a list of galleries that contain the image. You click the gallery name from which you wish to remove the image, then click the Delete Image button. The image will only be removed from the gallery you select. That will take a lot of time walking through each image.
The faster option is to display the contents of the gallery you wish to delete. (Again assuming the newer drag and drop editor) Click on the first image to select it. If you hold down the shift key and select the last image, select frames should appear around ALL of the images. Clicking the red 'X' tab should give you a pop-up "Are you sure you want to delete your selection of xx items? Yes No" (where xx is the number of images selected). Clicking 'Yes' will remove them from that gallery. You should then be able to go to the next gallery up ("the parent gallery"), go into edit mode and delete the empty gallery. You could try the block delete first with a block of three or four images, rather than all, to be sure you are getting what you want.
I am fairly sure the same operations can be accomplished with the old version of the editor also, but haven't done anything with it in a lo-o-ong time.
Hope that helps, the site is quite capable but not very user friendly.
DaveT
[PS: wrote this and when I went to post it, got a message that a message had come in since I started, etc. That bit about being logged out when submitting a forum post happens a lot. Normally just clicking the submit button again will make it go.]