It's sort of a hassle, but copy paste a block of text with a title, subtitle, header 1,2,3,4, etc into your old doc, then one by one, select a bit of the text with the correct style, and "update Title to match", "update Header 1 to match"...
And finally, select all > Format > Paragraph Styles > Options > Use my default styles.
Since Jul. 25th 2019 it's actually possible to specify in which account to create the new file, exactly as you stated in your question.
From now on:
docs.new/1 - creates the file in your personal account
docs.new/2 - creates the file in your second account (if you're signed in)
That applies to all .new shortcut URLs, check the explainer video from Google.
One workaround solution that hasn't been mentioned here is using Google's "Version History" feature to name a specific version. While that won't exactly lock your file down, it will give you an easy way to restore the version you want if it ever changes.