Identify that your cartridge needs FeRAM replacement. If the save game feature simply doesn't work, that's a good indication. 😉

Order all the parts needed. The FM16W08 chip costs around $11 at quantity 1 from most electronic part distributors at the time of writing. You can order the circuit board from your favorite fab house.  If this is OSH Park, you can order using the shared project.

Genesis cartridges are held together with some stupid security screws, and I've seen various ways of opening them ranging from melting pens to buying special screwdrivers. I find that just using a 4 mm hex driver and applying a lot of force, I can reliably open Genesis cartridges, so I recommend you try that if you already have a 4 mm hex driver.

Desolder the 24-pin FM1208S chip located at the top center of the circuit board. The holes aren't plated through, which makes the desoldering much easier than it could be.

Solder the new FM16W08 chip onto the adapter PCB. Set the solder bump jumpers to 0000 by bridging the bottom two pads on each jumper.

Put the adapter PCB flat against the cartridge PCB. Don't do anything to stand it off, because the cartridges are quite constrained in height. Put a bare 22 AWG or equivalent wire through a hole on both boards (preferably a corner). Keeping the two boards aligned, solder the wire to both.

Connect the holes at the opposite corner next. If the boards aren't properly aligned, reflow a joint made in the previous step and rotate the adapter board to the right angle. Once that's done, soldering all the other pins is easy, if a bit tedious.

Put the PCB back into the cartridge, then screw the two halves of the case back together.

Test the cartridge by selecting a save slot, then playing as many levels of Sonic 3 & Knuckles as you want (playing vanilla Sonic 3 is blasphemy). Resetting the Genesis, you should see the save list has one slot with a game in it. Congratulations, you fixed your Sonic 3 cartridge!

If you want to loan the cartridge to a friend, but don't want that friend to mess with your saved games, you can open the cartridge and change the solder bump jumpers to select a different block of memory. The friend will then have a completely new set of save slots to use, and when you get the cartridge back you can set the jumpers back to the original setting to get your saves back.