I just bought Digimon Story Cyber Sleuth physical copy a few days ago. If I buy a Digital copy to get the dlc, do I have to delete my saved game?
The saved game is stored on the hard drive, not on the physical disc. As long as you have a copy of the game (physical or digital) your save game will work. The only exception to this is in the case of a new edition of a game, but this occurs rarely outside of games with separate editions in separate global regions.
When you purchase the digital game, the PSN will recognize your license and turn your local phisical game copy into a digital one, without losing any data. Unfortunately, in some situations, your game might be "edition specific" (in the sense you will have to re-download it, and your saves might or might not work), but this thing happening is pretty rare.