I tried Dvorak for a while but gave up after I realized:

1. Not only are the key positions for typing different, but all the keyboard shortcuts I know needed to be re-learned too, and were often in much worse locations (as just one of many problems, Ctrl+C and Ctrl+V are no longer next to each other)

2. I still need to be good with QWERTY because basically everything uses it.

It's a shame, because it does seem more efficient (you can type a lot of words without having to move your fingers from the home row), but I just felt like it was too much work for too little gain.

It is one of the things I like about Colemak that it takes into account QWERTY's dominance in computers and intentionally doesn't move a lot a shortcut keys and punctuation. Semicolon is the most obvious punctuation mark that does move, most of the rest stay where they are. The big list of shortcut keys that stay the same place includes AZXCVBMWQ. XCV not moving is especially a big deal, as you also mention copy and paste as key shortcut examples.

In general Colemak doesn't move keys to radically far from their QWERTY positions (many stay on the same finger, even), so I've heard using it and QWERTY together isn't too bad. In my personal case, I had a terrible QWERTY form and realized I don't have to do anything but "hunt-and-peck" in QWERTY when using other people's machines. I may feel a bit like an idiot doing it, but it turns out, most people don't even notice if you touch type or hunt and peck (partly because so many more people hunt and peck than you tend to realize; even in a world where QWERTY touch typing has to be taught in schools, I think it is still a skill like Cursive that you think everyone else uses once you learn it, but you forget how hard it was to learn in the first place). It's more "my guilt" that I "use QWERTY poorly" than anything anyone else actually cares about.

Not only are Ctl+C and Ctl+V not next to each other, but they move next to other shortcuts that are annoying to mis-hit, like the one to close a browser tab...

Remapping the shortcuts is possible but too much effort; I just live with it.

Fortunately, modern video games seem to understand different layouts and automatically change their input to match the actual finger placement rather than what they QWERTY letters would be.

Re point 1, the standard solution is a keymap that switches to Qwerty when ctrl/command is held down. It works surprisingly well, and is widely available.

Re point 2, I have not found this to be true (when do I ever need to use someone else's keyboard?), but also I haven't lost my qwerty skill anyway.

For me it was the command line. 'ls' and 'cd' with slashes are exceedingly annoying to type often on a dvorak.