The other answers suggest using a built in "take a photo" in an app that also supports syncing. This is a smart way to do it. I'll also mention Bear, which is not free like Evernote, but relatively cheap and also supports this workflow.
To give more options for people who maybe don't want to do it that way for some reason... I did want to mention Workflow app, which would get you at least some of the way there no matter what apps you use (maybe).
You could make a workflow consisting of these actions:
- Take photo
When the workflow runs, it opens the camera, waits for you to take a photo and then moves on to this screen
From there, you could do many things: Copy to clipboard, save the image, upload to dropbox, airdop, add to various applications, etc.
The Share action was just a broader example... Some of the things you can do from that Share prompt, exist as actions in the Workflow app. Which means it's more automatic (or fully... it kind off depends) and you can chain them.
As an example, there is "Save File" action that you can use to save straight to Dropbox and iCloud Drive (which means you could even chain it and upload to both of them).
As it is, the workflow doesn't copy the photo to the clipboard or save it in the photos, but those actions could be added into the workflow.
Those two actions go by names: "Copy to Clipboard" and "Save to Photo Album". I've not used it, but these pre-made workflows say it supports universal clipboard..
The workflow could be triggered via: