I wish you a slow recovery. There is nothing you could say to your kids or spouse, been there after a while words are just noise to them . You will have enough time to wallow in the guilt and shame. The drink has to be put down. Actions not words are what will keep you sober. Zero desire that drink today... it will come back to you we have built in forgetters in ourselves. Find a program , I use AA it works for me.
Recovery is not easy especially in the beginning I wanted to drink everyday but the craves eases and finally subsides.
Saying I’m sorry it will never happen again never worked because I’m an alcoholic.
It sounds like you will have strong support from them , but that will take time