I am a divorced mother wishing to relocate overseas with the kids, who are in my custody. Will I be ‘stuck’ with rabbinical court (where I divorced several years ago) for the new, relocation proceedings, or can they be filed at the family court?

No, you are not ‘stuck’ with rabbinical court jurisdiction. Primary jurisdiction for relocation lies with the family court where the plea can be filed, directly.
Even if you expressly consented to rabbinical court jurisdiction at the time, children have independent legal personalities and are not bound by the choice/s of their parent/s regarding the judicial forum they chose to adjudicate on legal proceedings concerning them, according to long-standing Supreme Court of Justice binding precedent. Your children can initiate relocation proceedings themselves, as plaintiffs, via you, as their natural guardian, in the family court. You can also file for relocation, as a plaintiff, too.
Relocation From Israel