you want to delete files from you phone but the cloud copy will not be affected try to open Google Photos then slide from the left to right you will see all the menus when try to scroll down and see On Device menu that will open the images you have in your phone and when you delete images it won't affect the cloud files.

But how will I see the album organization on my phone?

One of my menu items in the Photos gallery is "albums" and I can see them from my phone. That said, I don't use the albums feature very often...what are you trying to do/see with albums?

What I would expect to see if a duplication of the albums setup on my phone - but when I click on the albums item, it searches then returns a 'no albums found' error.

What I would like to do is upload everything to the Google cloud as full size images, then delete images and videos from my phone to free up space, and then use the Google albums app to access my photos.

What I'm finding is the app did not upload the full size images, and it didn't retain the album settings either - so I just have a mass of photos jumbled together.

I believe the upload all go into the Autobackup album. You have to manually 'copy' them into albums. The process is MUCH easier/faster on a desktop browser. If you use Google+, there is a to do a search and upload all files on the phone. Photos app allows you to control which folders are auto-backed up (but you have to enable auto backup first, you also get to choose Full Resolution or free is a maximum of 2048 pixels on the larger dimension).

Deleting photos from Gallery does not remove from Autobackup (cloud).

Also, while albums may be visible in Gallery, you may only have thumbnails. If you want to make sure the full-resolution picture is available, you need to make an album 'Available Offline' then Gallery will download them.