New apple ID, rebuying apps without losing existing app data

I have been on my parents' apple ID for several years. I wanted my own separate Apple ID. When I made this new ID I ran into a problem. I don't own any of the apps on my phone with my new Apple ID. I tried to use a second device to purchase these apps so they were registered to my new account. This did not work. They are registered to my account but my iPhone did not care. I would delete all the apps from my phone and re-download, however, I have several apps with data that will be extremely annoying to lose. Does anyone know of a workaround?