If I go on a medication for hypothyroidism like Synthroid (thyroxine) to repair a short term issue secondary to other meds will I be dependent on it for life?

Need vs dependent. If one's doctor has recommended synthroid (thyroxine) for a hypothyroid condition, it is because of a need. Occasionally one may no longer need the synthroid (thyroxine) and it can be stopped without fear of dependence. Most, however, will need it lifelong.