OP, I understand your frustration. Having just been through this transition myself last week when I upgraded from the RAZR Maxx to the Droid Turbo.
My RAZR Maxx allowed me to create new albums/folders within the gallery and alas.. on the Turbo I cannot (at least after a week of searching/hunting/investigating).
Given your question, I know this isn't what you want to hear, but I found the app ES File Explorer (free on Google Play Store) a superb utility. Not only can I create my much needed gallery albums, but I can also manage my internal storage (ringtones, pictures, notification and alarm sounds, etc.). Since the Turbo has no microSD card, ES File Explorer gives me the ability to still have control over the device storage much like I had over the SD card.
Again, I'm sorry its not the inclusive on-the-phone method you were wanting and are used to, and I'm not making any promises, but I really think the ES app will serve you well.