Will deleting previous saved files allow some of the menu lag to be relieved or will deleting them remove my progress (which I absolutely do not want)?
Thanks for any help.

The only benefit I found in deleting old save files is it reduces the lag on the Save and Load menus (it has less to bring up). I'm not sure if this will change the lag when using other menus as I always seem to get it and have gotten used to tapping the joystick in the direction I want 3-4 times to make it move.

Lag is here to to stay.