Can I just replace the old mod and this mod keeps my old factories?
What about SE? I currently have a different mod for SE factories. Do I need to keep that one?

If you have an existing save you will have to install this version instead.
https://github.com/MagmaMcFry/Factorissimo2/pull/69

This version only has the electric network changes, but is compatible with existing saves.

Or just pick up all existing Factories and replace them after switching the mods, if it's not too much of a hassle

Depends on how many of the older factory buildings you have to replace, as you have to fully blueprint and deconstruct their interiors, and even deal with recursion too.

Or just pick up all existing Factories and replace them after switching the mods, if it's not too much of a hassle

This doesnt work. In my testing any old factory building when you place it down again causes it to disappear saying "Invalid factory data"

True, you don't get to keep the actual Factory items, but it means you can still use that save.