Go 2cm back. You might notice a difference but you will get used to it quickly. If you are really worried (and I think the Lizzy is symmetrical if I remember correctly) you might be able to get away with going 1cm from the previous holes if you want to maintain as much of the same feel as possible. I cant remember where but I remember a magazine or maybe it was Blister who did a test where they completely swiss cheesed a set of skis and had success with mounting holes as little as 3/4 cm from other holes. You may even be able to mount them in the same position if you can find a binding that the mount pattern is different enough that the holes don't overlap. You could print out some different mount templates to see what works best or find a good shop that will use a few different mounting jigs to help you decide what will work best.
Look Pivot/Rossignol FKS
I am a big fan of the Tyrolia Attack series bindings, other popular bindings are look pivots and the Salmon Sth2. Really you cant go too wrong with bindings unless you are getting the cheapest lightest ones. Attack 13 can be found for cheap and are branded as quite a few different bindings. The pivots have a really well designed heel and many people swear by them, but they can be finnicky to get on sometimes in pow. This is why I haven't been a fan in the past, but I also know a lot of people who don't mind this too much. I think the Pivot 14 would be a great choice. If you want a binding that will last you a really long time and you don't mind spending a little more, they are bringing back the 15 din model witch has the metal toe from the hailed Pivot 18 but with a lighter spring. Another benefit to the Pivot is that the heel mounting pattern is pretty far forward and compact relative to the Atomic mount pattern so this may be helpful for getting the
Pretty much anything will be an upgrade from the FFG12. Anything above a 12 din binding seems to be built with a little more durability in mind across the board for binding manufacturers. This is not directly related to DIN values to be clear, only an observation of pricing and build quality trends.