We are staying together for the children even though our marriage has long gone cold. Is this the right decision?

Staying for the children can be a noble sacrifice, but only if the home is still calm and safe for them. Children feel the atmosphere, not just their parents' marital status. A home that is cold but respectful is entirely different from a home full of shouting, sarcasm or cruelty. Before choosing, try to revive the relationship with a counsellor's help; many couples find a way when they talk with a third party. If you do stay, make it a conscious decision, not a prolonged punishment for yourself. If you choose to separate, that too can be done with kindness (ihsan) so the children do not become victims. See marriage check-up.