What you were told is correct in that neither benefit will be impacted by you taking these benefits. First of all, since you were born before 1954 it is smart of you to wait to delay your benefit until the age of 70.
Once you are eligible for retirement benefits (or decide to take them) on your own record, SSA will pay that amount first. If your ex-spouse's benefit is higher then you will receive an additional amount so that the combination of the benefits equals the higher amount. If your amount is higher, then you will not receive an additional amount.
(source: https://www.ssa.gov/benefits/retirement/planner/applying7.html#h4)
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