Thank you for the reply GamerMama.

My first thought is that you may be hitting some sort of slowdown, causing your inputs to not be read correctly. The next time you do a dungeon try lowering or disabling battle effects (Character Configuration > Control Settings > Character > Battle Effects) as well as lowering Draw Quality and Water refraction in the Display Settings (System Configuration > Display Settings). Lowering these may be less taxing on the PS4 (unfortunately, there isn't much you can lower on the PS4 besides these to reduce slowdown).

See if this affects the input lag you are experiencing.

We hope we can be of assistance in helping you enjoy the exciting world of Final Fantasy XIV!