Most probably, the safest method is to use an exchange that lists both coins, and has provisions to do the split for you, that is, to send the "unified old coins" to an address attached to you on the exchange.  The exchange will then give you IOU of both types of coins.  You can trade those (for instance, selling off your bitcoinx and buy more bitcoin, or selling off bitcoin to get more bitcoinx).  When you will withdraw the coins, they will be 1 type only.

But be sure you use an exchange that knows how to handle this.  Most big exchanges learned this during the ETH/ETC split.  Some made big mistakes.  Some stole the ETC of their customers, pretending they didn't "list" ETC.