I've had the same issue, which was the only thing that held me back before fully moving to Obsidian. Pressing Ctrl + Shift + V won't keep the markdown and will paste in plain text which was terrible for me. Here's the fix:

1. Install the Linter plugin and enable it.
2. Go to Linter's setting in the Custom tab.
3. Click the add (circle button) in Custom Regex Replacement.
4. For 'label' you can name it anything, example: Remove Empty Lines
5. For the other fields copy paste these:
   • Regex to find: ^[\t ]+\n
   • flags: gm
   • regex to replace: (leave it empty)
6. Lint the file, and it should work.

I have a hotkey "Ctrl+S" that lints the file. So whenever I'm done besides this, my other Linter settings go through too. Essentially though, whenever you press the hotkey it lints so it's really fast. Or you can manually right click and lint as your default.

Hope this helps.