On PC with the controller UI (Xbox gamepad), Y is used to change profile in the main menu, so on PS4 I'd guess the triangle button. The profile and button to change it should be shown on the left edge of the screen, near the top, rather than in the list with the other main menu options.
If you load the new game again, can you make manual saves (so you are not overwriting quick or auto saves)? Then load your current save and see if you can make manual saves (so overwriting an existing save isn't an issue).
Can you load a current save, then load it again, or return to the main menu?
Please email firstname.lastname@example.org, with a brief description (or link to your post) and one of your recent saves (or a Dropbox or Google drive link, etc). You would need to transfer it to a USB flash drive first, to get it onto a computer for that. In the Saved Data in System Storage section of the Settings menu, click Copy to USB Storage Device, select the relevant save and Copy.