I moved my presale ether to another wallet that has never made a
transaction from it. As I understand, that wallet now exists on 2
chains (eth and etc) with identical private and public keys.
You should check blockchain explorers to find out which address your ETH is on, and which one has the ETC. It depends on how you "moved my presale ether to another wallet", which chain you used, and if it was replayed on the other chain.
Is my ether safe if I don't touch it?
What do I do about my ETC?
Your choice and ETC pays for gas on the ETC chain.
Do I need to put my ether through a splitter contract
Not necessarily. If you use a new client, such as Geth 1.5.3, it implements EIP 155 Simple replay attack protection so that your ETH transaction should be safe from a replay attack on ETC. You probably should move all your ETH to another address using Geth 1.5.3, then afterwards on the ETC chain move all ETC to a different address. This will help ensure that you are indeed protected from replay attacks.