Depending on your electrical installation inside the walls your #2&3 may or may not work. If there’s a metal conduit used to distribute wiring through the house this would sufficient to ground all of your new 3 prong receptacles.
Since some jurisdictions require extra grounding (bonding) between the receptacle ground screw and the electrical metal box you may need to install it while changing the receptacles, but, if there is no metal conduit (or 3rd wire) between the box and grounded main electrical panel, the ground wire (between the receptacle and the electrical box) will not provide any extra protection.
Installing GFCI breakers for all of the house’s electrical circuits (those breakers are expensive) will protect you from being electrocuted but it will not provide ground for the equipment plugged into it.
So, the solution would depend on your wiring. Without a metal conductor (3rd wire or metal conduit) between the outlet boxes and the main panel you can use GFCI breakers or GFCI outlets to protect each circuit. If there’s a metal conduit all you need to do is replace 2 prong with 3 prong and add an extra wire inside the box if such is required.