Have I fallen short in anything, and what is the ruling of Sharia on giving zakat from the money saved in my two savings accounts to my twin daughters, knowing that I spend from it on my family when needed?

Depositing money in a savings account falls under the category of Riba (usury), and one must repent from it and dispose of the usurious gains by spending them on charitable causes. What you have taken from it previously out of need is permissible. The money gifted to your two daughters is considered their property, and it is permissible for you to take from it according to your need. Zakat is obligatory on a minor's wealth if it reaches the Nisab (minimum threshold) and a full year has passed on it. One-quarter of one-tenth (2.5%) must be paid for each year separately. You are not considered to be neglecting your children's rights as long as you fulfill their rights regarding sustenance, clothing, and education.