Am I Understanding Form 709 And The Gift Tax Rules Correctly?

My wife and I gifted jointly $158,000 of value to our son and daughter-in-law in 2022.

My understanding is that my wife and I must separately file our own form 709 showing our share of the total gift.  The 2022 gift amount would be $79,000 from each of us.   Both recipients can be on each of our individual form 709.  For tax year 2022 my wife and I each are allowed a $32,000 exclusion.  The balance of $47,000 for each of us can be applied to our individual lifetime exclusion.   We have never done this before so there is no carryforward.  So far, am I on track?  Any thoughts or suggestions will be greatly appreciated.