Ah, the old “It won’t save anymore!” problem. Yes, I’ve had this one. When I first bought Shining Force 1, it was a used copy of the game and after a day of solid playing and getting most of the way through the game, I switched off, only to return the next day and find nothing!
You should be able to take it into a games store and they can do it for you (that’s the expensive option).
The way I got around it was to carefully unscrew the back panel of the cartridge (usually this involves heavily mangling the plastic screws in the back) and removing the offending item (dead battery). Then, you need to take it to a battery-selling store and find one that’s the same (if you’re not sure, ask the guy at the counter for help – you need to get the right one). Take it home, put it in the cart the same way around as the old one was and put the cart back together. You may well have to use new screws in it, as the old ones are probably beyond repair by this stage. Try saving a game after this and all should be well.
If you’re going to open the cartridge up yourself to replace the battery, you can actually buy a tool to remove the screws without damaging them (if you’re in the UK, click here).
I must say however that if anything goes wrong (unlikely) I can’t accept responsibility for it. If in doubt, get it done by a pro.