I think the whole trick usually is the reverse. Remove of Act.dat or whatever to make the microSD card "clean" with just games and put it into new PS Vita / PS TV that is SIGNED to an account on Sony part and then the microSD cards gets connected properly and the games can be played again.
PS Vita saves are somewhat encrypted I believe, putting aside for a moment what an emulator can do or not, technically you can't just copy them to the emulator because "no key" to open it. PSP on that matter didn't have such a thing I believe, that's why you could just copy save to the emu or even to other PSP and continue playing.
If you want to move to new PS Vita without signing in to the PSN then I'm not sure if you can keep the saves but I guess you could try with the save backup tool. Of course make sure you have your own backup of the microSD card before proceeding just in case. The steps would be to backup the saves with the tool, remove Act.dat or whatever the files "hold" microSD in place, put it into new PS Vita, restore backups with the tool and see if it will work out.
Other than that you would have to somehow unecrypt the files - maybe with VitaShell ? - and then "re-encrypt" them for new PS Vita but you would have to somehow know what "signature" it is expecting.